How important is patience in buddhism?

From Dhammapada 184:

The best moral practice is patience and forbearance;
"Nibbana is Supreme", said the Buddhas.
A bhikkhu does not harm others;
one who harms others is not a bhikkhu.

Alternative translation from here:

Enduring patience is the highest austerity.
"Nibbana is supreme," say the Buddhas.
He is not a true monk who harms another,
nor a true renunciate who oppresses others.

The Pali verse for the first line is "khanti paramam tapo titikkha".

So, the word for patience here is khanti.

The PTS dictionary entry for khanti states:

Khanti & Khantī (p. 232) Khanti & Khantī Khanti & Khantī f. [Sk. kṣānti] patience, forbearance, forgiveness. Def. at Dhs 1341: khantī khamanatā adhivāsanatā acaṇḍikkaŋ anasuropo attamanatā cittassa. Most frequent combinations: with mettā (love) (see below); -- titikkhā (forbearance): khantī paramaŋ tapo titikkhā nibbānaŋ paramaŋ vadanti Buddhā Dh 184

One of the three basics to get useful answers: Respect, Confidence and Patient and hardly the enviroment here.

[Note that this isn't given for exchange, trade, stacks, entertaining bonds, but to escape from this world]

Restraint

I think that another way to frame patience is in terms of restraint. Why is this? If we see a mental defilement arise, we exercise restraint to not immediately give into it. We look at the situation and see it for what it is. In this way, we are exercising patience, but also restraint.

Now what does the Buddha think of restraint? Well, Buddhist monks (Bhikkus) follows the Dhamma-Vinaya, which is a set of 227 rules that they follow. This give us some hint of what the Buddha thought of discipline (read restraint and patience) on the path.

1 And what are the effluents to be abandoned by restraining? There is the case where a monk, reflecting appropriately, dwells restrained with the restraint of the eye-faculty. The effluents, vexation, or fever that would arise if he were to dwell unrestrained with the restraint of the eye-faculty do not arise for him when he dwells restrained with the restraint of the eye-faculty. “Reflecting appropriately, he dwells restrained with the restraint of the ear-faculty .… “Reflecting appropriately, he dwells restrained with the restraint of the nose-faculty .… “Reflecting appropriately, he dwells restrained with the restraint of the tongue-faculty .… “Reflecting appropriately, he dwells restrained with the restraint of the body-faculty .… “Reflecting appropriately, he dwells restrained with the restraint of the intellect-faculty. The effluents, vexation, or fever that would arise if he were to dwell unrestrained with the restraint of the intellect-faculty do not arise for him when he dwells restrained with the restraint of the intellect-faculty. These are called the effluents to be abandoned by restraining.

• MN 2

There are many a talks on Restraint, namely I think of this one by Thanissaro bhikku. There is a lot to say, but I will leave this talk for you to read. If you wish for more, I can further my discussion.

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Patience

On second thought, I have decied it is wise to discuss Patience in more detail. There is a relation between restraint and patience, but they are not the same.

“Endowed with these six qualities, a person is incapable of alighting on the lawfulness, the rightness of skillful qualities even when listening to the true Dhamma. Which six?

“When the Dhamma & Vinaya declared by the Tathāgata is being taught, he does not listen well, does not give ear, does not apply his mind to gnosis, grabs hold of what is worthless, rejects what is worthwhile, and is not endowed with the patience [or: preference] to comply with the teaching. ... “When the Dhamma & Vinaya declared by the Tathāgata is being taught, he listens well, gives ear, applies his mind to gnosis, rejects what is worthless, grabs hold of what is worthwhile, and is endowed with the patience [or: preference] to comply with the teaching.

“Endowed with these six qualities, a person is capable of alighting on the lawfulness, the rightness of skillful qualities even while listening to the true Dhamma.”

• AN 6:88

So, we can see that the Buddha put patience high on the list of what one must cultivate with respect to the Dhamma. Now what may we do with this? How may we practice with patience? In my practice, this primarily arises when looking at foundations of concentration -- ardency, mindfulness, and alertness.

We must be ardent in coming back to present phenomenon. That is that we must have patience in watching the mind. If we do not keep coming back to present phenomenon because we are not patient, then we have no hope of seeing phenomenon as they are.

“And what is the faculty of sati? There is the case where a monk, a disciple of the noble ones, is mindful, highly meticulous, remembering & able to call to mind even things that were done & said long ago. (And here begins the satipatthana formula:) He remains focused on the body in & of itself — ardent, alert, & mindful — putting aside greed & distress with reference to the world. He remains focused on feelings in & of themselves... the mind in & of itself... mental qualities in & of themselves — ardent, alert, & mindful — putting aside greed & distress with reference to the world.” - SN48:10

Here is a talk on patience by Thanissaro Bhikku. Another talk by Gil Fronsdal.

I hope that you are able to be ardent and patient in your daily practice. Watch the phenomenon and see how it affects the mind and body. Patience is a must on this path.

Khanti-Parami

Patience is the unimpeded weapon of the good in the development of noble qualities, for it dispels, without residue, anger, the opposite of all such qualities. It is the adornment of those capable of vanquishing the foe; the strength of recluses and Brahmins; a stream of water extinguishing the fire of anger; a mantra for quelling the poisonous speech of evil people; the supreme source of constancy in those established in restraint. Patience is an ocean on account of its depth; a shore on account of bounding the great ocean of hatred.

Patience should be fortified by reflection: “All beings are like my own children; who becomes angry over the misdeeds of his or her own children?” Or, “I am wronged by others because of some residue of anger in myself; this residue I should remove.” Or, “A wrong-doer is a benefactor, for he or she is the basis for developing patience.” Or, “If there were no wrong-doers, how could I accomplish theperfection of patience?”

“When there is patience, the mind becomes concentrated, free from external distraction. With the mind concentrated, all formations appear as impermanent, stressful, and not-self. In addition,Nirvanaappears as unconditioned, deathless, peaceful, and sublime.The groundlessness of “I-making” and “mine-making” becomes evident to reflection thus: ‘Mere phenomena alone exist, devoid of self or of anything pertaining to a self; they arise and pass away in accordance to their conditions. They do not come from anywhere, they do not go anywhere, they are not established anywhere. There is no agency in anything whatsoever.’

• Dhammapala's Treatise on the Paramis

Edit: I forgot to add that patience is one of the ten paramis. This is another discussion and avenue to look into this concept. Here are some quotes from suttas regarding Khanti-parami (The perfection of patience). Here is a book on the paramis

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With regards to your last question, I think it's important not to frame questions in this way. "Does the person have excessive clinging?" Clinging is present in the mind, but a person does not have excess clinging. Clinging is an ongoing process. Regardless of where someone is on the path, clinging will arise and cease. Sometimes there will be more or less, but it is not an absolute indicator of where one is along the path. See clinging as the arising and ceasing of phenomenon -- in terms of anattā. See it in terms dependent origination (paṭiccasamuppāda) -- clinging arises as a consequence of something and passes away as a consequence of something.

Something that has helped me is to frame a clinging not as a negative (e.g. I'm not far on the path), but a positive. It gives one a chance to see it in terms of the Dhamma. Everytime you notice a clinging, you are making progress (if dealt with correctly). See it arise, notice your bodily interactions and release them. See the defilements like to sneak in.

Please let me know if you have further questions or would like additional references.

Metta

And what’s the impatient practice? It’s when someone abuses, annoys, or argues with you, and you abuse, annoy, or argue right back at them. This is called the impatient practice.

And what’s the patient practice? It’s when someone abuses, annoys, or argues with you, and you don’t abuse, annoy, or argue back at them. This is called the patient practice. An4.164

And what’s the impatient practice? It’s when a mendicant cannot endure cold, heat, hunger, and thirst. They cannot endure the touch of flies, mosquitoes, wind, sun, and reptiles. They cannot endure rude and unwelcome criticism. And they cannot put up with physical pain—sharp, severe, acute, unpleasant, disagreeable, and life-threatening. This is called the impatient practice.

And what’s the patient practice? It’s when a mendicant endures cold, heat, hunger, and thirst. They endure the touch of flies, mosquitoes, wind, sun, and reptiles. They endure rude and unwelcome criticism. And they put up with physical pain—sharp, severe, acute, unpleasant, disagreeable, and life-threatening. This is called the patient practice.an4.165

What taints, bhikkhus, should be abandoned by enduring? Here a bhikkhu, reflecting wisely, bears cold and heat, hunger and thirst, and contact with gadflies, mosquitoes, wind, the sun, and creeping things; he endures ill-spoken, unwelcome words and arisen bodily feelings that are painful, racking, sharp, piercing, disagreeable, distressing, and menacing to life. While taints, vexation, and fever might arise in one who does not endure such things, there are no taints, vexation, or fever in one who endures them. These are called the taints that should be abandoned by enduring.mn2

One who bears, without resentment; verbal abuse, assault & captivity; with forbearance as his strength and mighty army, I call the Foremost[brāhmaṇaṃ] (It's my amateur translation)dhammapada, verse 399

I think these are self-explanatory

Khanti/patience is indeed a very important virtue to cultivate. It's listed in Sn 2.4 as among one of the greatest protection for a practitioner. Also refer to many other related suttas.

Patience, compliance, seeing contemplatives, discussing the Dhamma on timely occasions: This is the highest protection. ~~ Snp 2.4 ~~

Something Eckhart Tolle once said... Patience is itself an expectation. When we are being patient, it effectively means that we are experiencing something we dislike, but have decided to wait calmly in the expectation the experience will eventually dissipate. Being patient in this sense is effortful, and like all efforts it will eventually run out of steam. If the disliked experience doesn't dissipate quickly enough, we will lose patience, with all the unfortunate consequences that has.

This isn't to say that patience is bad. Patience is right effort. But we don't want to root our practice in patience, because that will ultimately lead us around in circles.