Has there ever been a cold war other than between the U.S. and the U.S.S.R.?

From the earliest time of the very first ever peace treaty that we know of, we also have a 'cold war':

In the beginning, both sides seesawed violently between victory and defeat, before settling down to an inconclusive tussle in which nobody won very much for long. Hatti, losing central Syria to Sety, got it back after the Battle of Kadesh. Ramesses II lost Upe in the wake of his Kadesh campaign, but regained it subsequently. Later efforts to improve on these positions were unsuccessful, and in the end both sides lapsed into about a decade of cold war.
_{Summarized in K. A. Kitchen: "Pharaoh Triumphant: The Life and Times of Ramesses II, King of Egypt", Warminster, 1982, pp. 62–75.}
[…]
To get rid of Aziru would then oblige Egypt to interfere directly in Amurru, to extend her own military establishment beyond its preferred limits with unpredictable results, perhaps even the breakup of Amurru and still more defections to Hatti. To trust Aziru was to gamble on the one man who had shown himself resourceful enough to unite Amurru under his rule. Egypt chose Aziru. If the Pharaoh meant to enjoy the advantages of relying on a powerful proxy, he now had little choice but to countenance a strong kingdom of Amurru; and Aziru, now that Hatti was in league with his enemies, had every reason to ally himself with a power whose interests marched with his own.
[…]
The invasion of Nuhashshe belongs to one of the war's later phases, as we have seen; and these letters of Aziru should have been written after EA 55, which shows Aziru acting, in all probability, as a Hittite proxy before Shuppiluliuma had subdued Nuhashshe and before Aziru himself had been checked by the onerous presence of an Egyptian watchdog.
[…]
In any case, having ranged himself with Shuppiluliuma's enemies, the pro-Mitannian factions in Syria, Shutatarra was treated accordingly. By putting himself in the way of the Hittite juggernaut he had, in effect, acted as a partisan of the Mitannian king. His behavior gave the Hittites a reason to justify their continued sway over Kadesh in later years, when Aitakama was their willing proxy in Syria: Carchemish, Shuppiluliuma breezily informed the Egyptian envoys that he had taken Kadesh away from the Hurrian king--for, having behaved as if she were a Mitannian vassal at that time, this was de facto what Kadesh had been, and her entry into the Hittite orbit was none of Egypt's business. This was a fiction that the Hittites would feel obliged to maintain for as long as they felt the need to make a case against Egypt's prior claim on Kadesh. Later on, with the city in Hittite hands and Egypt estranged from Hatti, it could be safely admitted that, in taking Kadesh, Shuppiluliuma had inflicted a defeat on the king of Egypt.

_{— William J. Murnane: "The Road To Kadesh. A Historical Interpretation of The Battle Reliefs of King Sety I at Karnak", The Oriental Institute of the University Of Chicago, Studies in Ancient Oriental Civilization, No. 42, The University Of Chicago, Chicago, 1985.}

Having a 'cold war' in general is seen quite often, and once the term for this was invented, historians applied this very liberally to characterise such situations in the past. But 'having a 'cold war' and really 'hot wars by proxy' mixed in at the same time is usually a situation that has to be described as 'unstable' or 'escalating quickly'? When to draw the line between hostile non-fighting, aggressive interludes of 'peace', and all out hotness may not always be so easy if we don't introduce an essentially arbitrary cutoff for the length of time for each such a phase.

Such an example might be found in all the little preludes and sideshows of the Second Punic War, one of them being Macedonian—Carthaginian Treaty:

The Macedonian–Carthaginian Treaty was an anti-Roman treaty between Philip V of Macedon and Hannibal, leader of the Carthaginians, which was drawn up after the Battle of Cannae when Hannibal seemed poised to conquer Rome. Philip V, who feared Roman expansion, wanted to ride on the coat tails of the victor in the Second Punic War (218–201 BC). The discovery of this treaty inevitably led to the outbreak of the First Macedonian War (214–205 BC) between Rome and its Greek allies against Macedonia.

In the same manner we see then the description of

The Seleucid War (192–188 BC), also known as the War of Antiochos or the Syrian War, was a military conflict between two coalitions led by the Roman Republic and the Seleucid Empire. The fighting took place in modern day southern Greece, the Aegean Sea and Asia Minor.

The war was the consequence of a "cold war" between both powers, which had started in 196 BC. In this period Romans and Seleucids had tried to settle spheres of influence by making alliances with the small Greek city states.

At any time, between most countries according to your definition.

At any given point in time Country A is in state of conflict with other nations. Country A is currently

engaging in economic and political actions, propaganda, acts of espionage or proxy wars waged by surrogates

with Country B. Country B is aware of that, and does the same with Country A, as well as with Country C, D, E etc.etc.

The definition you found describes well US and China, or US and Europe, or UK and Europe on trade wars. Or Russia and US on geopolitical issues. And many more.

Or to use some famous quote - according to Carl von Clausewitz, a Prussian general and military theorist of the 19th century: "War is the continuation of politics by other means".

To summarise: according to the definition you found, cold war is the normal state of actions between any two countries. I would specify "open state of conflict" in your definition - otherwise your cold war is no different from Clausewitz peace.

It is possible that there was was something like a cold war in the Middle East between the Roman empire and the Parthian empire of the Arsacid dynasty succeeded by the Persian empire of the Sassanid dynasty for about 700 years.

There were many wars and many periods of peace in those 700 or so years but it is possible that there were periods of cold war and sometimes proxy wars by vassal or allied states during those periods of peace.

I may note that Harry Turtledove wrote a series of stories collected in Agent of Byzantium (1987) set in an alternate universe where Mohammed converted to Christianity and never founded Islam, and so the Roman and Sassanid Persian empires still continue their rivalry and "cold war" like actions in the Middle East in the 14th century.

Cold War has been used to describe the current tension between China and India by people such as Imran Ali Sandano of the University of Sindh, journalist Bertil Lintner, and politician-lawyer P. Chidambaram. Tense relations between China and India have led to three military conflicts: the Sino-Indian War of 1962, the Chola incident in 1967, and the 1987 Sino-Indian skirmish. There was the 2017 clash over the Sino-Bhutanese border and there are the recent border skirmishes that go on to this day.

Given all the answers above we certainly can find more cases, but I'll give a current emerging example:

The China - U.S. cold war (for Africa)

At the 12th US-Africa Business Summit held in Maputo, Mozambique there were 11 African heads of state and government and around 1,000 business leaders. During the event, US officials unveiled that a $60 billion investment agency which seeks to invest in low and middle-income countries, with a very special focus on Africa.

6 months before that, National Security Advisor John Bolton presented to the US administration what he called "The New Africa Strategy".

According to the presentation, "Great power competitors, namely China and Russia, are rapidly expanding their financial and political influence across Africa. They are deliberately and aggressively targeting their investments in the region to gain a competitive advantage over the United States."

While both are mentioned, the focus in most of the meetings and debates that happened recently is on China. My many Africa is now the next possible big battleground for the escalating trade war between Beijing and Washington.

Africa at this moment is witnessing the first signs of an emerging new cold war. The obvious signs of that are the ever-increasing foreign military presence, planned investments and general diplomatic tension. This is nothing new to Africa and clearly something detrimental to African development and peace.

Economic wars

China's approach to Africa was and is trade-focused oriented. Africa became one of the top destinations for Chinese investment after Beijing introduced the so-called "Go Out" policy (Zǒuchūqū Zhànlüè) in 1999. This policy encouraged private and state-owned business to seek economic opportunities abroad. And they succeeded both in Europe and in Africa, but in the case of Africa the media coverage is a lot lower. The practical result: Chinese trade with Africa has increased 40-fold over the past two decades. In 2017 it reached $140 billion. Between 2003 and 2017, Chinese foreign direct investment (FDI) funds have increased more close to 60-fold to $4 billion / year. FDI stocks stand now in 2019 at $43 billion and those are practical things because a significant part of that has gone to infrastructure and energy projects: expansion of African railways, various infrastructure projects in Angola (among others: hydro-power plant under construction), Djibouti, Ethiopia (Africa's longest railway between these last 2), Kenya and Nigeria.

US, by contrast, has viewed Africa as a battlefield where it can confront its enemies, whether the Soviets, 'terrorists' or now the Chinese; while that they did not focus on developing serious economic relations. As a result, trade between the US and Africa has decreased from $120 billion in 2012 to just over $50 billion today in 2019. So the US it will not be able to challenge Chinese economic presence on the continent. In 2018 Chinese President Xi Jinping pledged $60 billion more be invested in Africa.

The US has continuously accused China of using "debt to hold states in Africa captive to wishes and demands" while propaganda-warning the African nations of "a significant threat to US national security interests".

Africa started to own China, currently that being around $83 billion.

To make the situation clear, Djibouti's President Ismail Omar Guelleh has stated: "The reality is that no one but the Chinese offers a long-term partnership."

The pressure the US is currently exerting on African countries to move away from partnerships with China could hurt African economies. It could force African countries into making choices that are not in their best economic interests and miss out on important development projects or funding. While something like this should not happen, history tells us that it practically does almost every time.

The current US-China trade war is already affecting Africa, as is everyone else. According to the African Development Bank, it could cause as much as a 2.5 percent decrease in GDP for resource-intensive African economies and a 1.9 percent dip for oil-exporting countries.

Militarization

Both US and China are militarily involved in Africa. The escalating tensions between them could end up threatening the security of the whole continent.

Over the past 15 years, the Chinese People's Liberation Army has been engaged in a number of security missions across the continent, making modest auxiliary troop contributions to peacekeeping operations in the Democratic Republic of Congo, Liberia, Mali and Sudan. It has also contributed millions of $ of peacekeeping equipment to the African Union Mission in Somalia and provided significant funding to the Intergovernmental Authority on Development for its mediation in South Sudan.

In 2017, the first Chinese overseas military base was opened in Djibouti. The facility has the capacity to accommodate 10,000 troops and officially it is supposed to provide support for the ongoing anti-piracy operations of the Chinese navy (maritime routes security). Rumors say that this is only the 1st base in many.

But as said above, China focused on the Economy part, while US focused on the military presence. Over the past few years, US Africa Command has run some 36 different military operations in 13 African countries, including Burkina Faso, Cameroon, Central African Republic, Chad, Democratic Republic of Congo, Kenya, Libya, Mali, Mauritania, Niger, Somalia, South Sudan and Tunisia. It has more than 7,000 troops deployed on the continent. This may not seem much, but the operations done there are quite significant if seen in an over-all manner.

US has a main base in Djibouti and at least 34 other military outposts scattered across the continent. Various military operations including drone attacks and sabotage are launched from all these outposts. The US directly supports the armies of Egypt, Nigeria, Ethiopia, Mali, Niger and others as well as the G5 Sahel force tasked with 'counter-terrorism'. This type of operations become an increasingly destabilizing factor. Practically, US counts on the conflicts and social unrest to counter the Chinese influence in Africa.

Regional tensions

Djibouti has recently found itself at the centre of US-Chinese diplomatic war. Being a host to military bases of both superpowers, they are forced o tr to keep things in balance.

In 2018, Djibouti seized control of its Doraleh Container Terminal from the Emirati company DP World, claiming its operation of the facility was threatening its sovereignty. The Djibouti authorities had feared that the UAE's investment in the nearby Port of Berbera in the autonomous Somali region of Somaliland could challenge its position as the main maritime hub for Ethiopia's large economy.

Its decision to terminate the contract with DP World triggered a sharp reaction from Washington, a close Emirati ally. The US administration fears that Djibouti could hand over control of the terminal to China.

"Should this occur, the balance of power in the Horn of Africa - astride major arteries of maritime trade between Europe, the Middle East, and South Asia - would shift in favour of China. And, our US military personnel at Camp Lemonnier could face even further challenges in their efforts to protect the American people", National Security Advisor John Bolton warns.

Djibouti was forced to declare publicly that it would not allow China to take over the terminal.Even so, ever since, the US sought to secure a possible alternative location for its African military base in Eritrea. Saudi Arabia and the UAE were encouraged to pull Eritrea out of its decades-long isolation. In a matter of months, long-time enemies Ethiopia and Eritrea concluded a peace agreement to end their 20-year-old cold conflict and the UN lifted sanctions on Asmara. As a result, Eritrea could emerge as a strategic rival to Djibouti, offering its coast for foreign military and economic facilities. The UAE, for example, has already set up a military base near the port of Assab. If this is not cold war tactic, what is ?

Another area that has been an unofficial battleground is Sudan. China had been a long-term supporter of President Omar al-Bashir. Under his rule, Beijing came to dominate its oil industry, buying some 80 percent of its oil and thus providing Khartoum with much-needed cash to wage war against various rebel groups (part of which had various levels of US-support). After South Sudan gained independence in 2011, China continued to be a close partner of the Sudanese regime, remaining its main trading partner. Sudan in fact became the biggest beneficiary of the African investment package China pledged in 2018, and were spared about $10 billion debt. When mass protests erupted in December last year, Beijing stood by al-Bashir, who it saw as the main guarantor of stability in the country, which falls on strategic routes, part of its Belt and Road Initiative.

Meanwhile, the US had repeatedly demonstrated that it did not want al-Bashir running for another term. His removal was approved in Washington, which has since appeared to back the interests of Saudi Arabia and the UAE in the country. This is a full on-going operation of which results / effects we will soon see.

This political confrontations have added to the already rising tensions between other players in the region, including Egypt, Gulf countries, Iran and Turkey. The US administration has particularly favored Emirati, Saudi and Egyptian interests which have emboldened these three countries in their efforts to shape regional dynamics to their advantage.

Thus, in the long-term, given the pre-existing faultiness and conflicts in the region, the China-US cold war, if continued, will definitely have a detrimental effect, not only on Africa's economy but also on its security.

Russia vs Ottoman Empire

Throughout the period of the Russo-Turkish wars the times of "peace" would be better characterized as "cold war".

The period involved intense rivalry and border skirmishes and the sponsoring of raids on the other's territory by the local cossacks. This culminated in Russian support of secession movements and rebellions in the Balkans and ultimately the First World War: which proved to be the end of both states as they were then structured - Russia became the USSR and the Ottomans became Turkey.

(Nazi) Germany - Soviet Union

It would be hard to pinpoint the beginning of the "cold war" since the Soviet Union was actively supporting left-wing fractions during the Weimar Republic, but of course, those countries became even more hostile after Hitler became the chancellor. After all, one of the points of the Nazi party manifesto was "fighting communist plague" while the Red Army was ready to "send friendly help" to the oppressed workers in Germany. Both sides were using an extensive network of spies, although it seems that Russians were more successful in planting their agents in Germany than another way around.

The culmination of the "cold" part of this war was the Spanish civil war (1936-1939), in which Germany was supporting general Franco, while the Soviet Union was financing the Republican forces.

I consider the conflict between Sunni and Shiite Islam as a current "Cold War". Saudi Arabia considers themselves the leaders of Sunni Islam, while Iran considers themselves the people who speak for Shia. The current conflict among Muslim powers is substantially similar to the cause of the Thirty Years War.

Consequently, when any conflict between Sunni and Shia groups arise, Iran and Saudi Arabia aid their fellow co-religionists. In Yemen, for example, the Houthi rebels are predominantly Shiite (and the current conflict started over a proposal to gerrymander the nation into new regions that would have locked the Shia out of any political power). As a result, the Saudis assist the (officially) governing side of the civil war, while the Iranians assist the Houthi side.

During the "Arab Spring", the Sunni government of Bahrain invited the Saudis to send troops to quash the majority Shia from uprising. Generally, during the Arab Spring, the US promoted all of the rebels, except when the rebels were Shia.