How should I express the end of a period of time I'm allowed to stay in a country so as not to confuse it with visa expiration?

Visa duration or length of stay.

For example, for the US (while ESTA isn't a visa, exactly), an ESTA is valid for two years. However, you only get allowed up to three months in the country at a time in the US.

The US Embassy describes it well, and uses 'length of stay':

As example of the difference between the duration of stay permitted in the U.S. and validity of a visa, your visa may be valid for several years, and yet your authorized period of stay, as shown on the Arrival-Departure Record, Form I-94, may be limited to a few weeks.

Japanese terminology is similar to US one. When you show up at the border, the immigration official checks your paperwork (passport, disembarkation card, visa if one is required), and if everything is in order he or she will grant you a status of residence (or just status for short). Your status of residence has an expiration date, which is the date by which you must leave Japan (unless you apply for an extension, of course) and is indicated on the landing permission stamp affixed on your passport, as well as on your residence card if you have one.

As usual, the UK has their own terms for these types of things. Relaxed's answer has introduced the notion of 'leave to enter', which is what the Immigration Officer grants at a port of entry. It is contrasted with 'leave to remain', which is something a qualified person can get from inside the UK (e.g., 'further leave to remain' and 'indefinite leave to remain').

For your question about the number of days, there are two special terms:

• Unspent leave, the amount of time a person has before the leave they have been granted is depleted
• Spent leave, the amount of time a person has used against their initial grant of leave.

If you use these terms with an immigration official, they will know what you are talking about due to their training. You will also see the terms used in formal correspondence between lawyers and the Home Office...

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'Leave to enter', 'leave to remain', 'spent leave', 'unspent leave'... Normal people are not required to know these terms, and I have seen no evidence that people have any understanding of their meaning. They will use things like 'interval' or 'duration' or 'number of days until expiry' or related terms. For ordinary purposes there is no reason not to use the terms you are comfortable with.

Note: You can search Google for 'uk spent leave rule' and come up with a few hits of how it entered common parlance at one point (note: the 'spent leave rule' was a policy, not a rule; and it was abolished a long time ago). You may also find sporadic references in the Hansard for all these terms, but again this is elevated language and not usually targeted for mass consumption.

Note: An entry clearance contains fields for a start date, end date, and number of days. But the number of days field will contain the date difference between the start date and end date because the machine calculates it rather than a person. If they want to restrict a person's leave to a specific number of days, like 15, they will use the start and end dates to do it. In keeping with the terminology theme: your unspent leave will be depleted on your visa's expiry date.

Note: Schengen uses a different scheme where the start/end dates may surround a limited number of days. Sometimes I will resort to using the UK's terminology to describe how it works, but this is a one-off contextual usage and not part of Schengen terminology.

See also: Are "tourist visa" and "visitor visa" two terms for the same thing?

The US terminology is “status”, I have read things like “having status” and “duration of status” so the last day you would be allowed to stay in the US would generally be the “last day of status” or perhaps the day your “status expires” or something like that (with some caveats, there are also grace periods and situations in which you are allowed to stay without status, e.g. with a pending change or adjustment of status or with an asylum application).

The UK terminology is “leave to enter”. You are allowed to stay in the UK for the length of time set when you are granted “leave to enter”. Researching this answer, I have come across things like “when the leave to enter expires” (Ireland) or “the end of the period of leave” (UK) to describe the end of the period of time a person is allowed to stay in the country.

I doubt any of this is particularly clear to anybody who isn't already well-versed in all the subtleties of visa regulations.

Schengen visas do not work in the same way but the relevant regulations use a terminology that might be useful (if a bit clumsy, as is often the case with EU terminology): “maximum duration of authorised stay”. In the Schengen area, you are not allowed to stay beyond the expiration of the visa (unless you get another visa) but the phrase could perhaps also be used where and when that's the case.

(Also, in case someone is wondering, there is in fact a country where English is an official language that is part of the Schengen area: Malta. Incidentally, British or Irish EU civil servants, including translators, are involved in all the work related to the Schengen area, even if their country isn't part of it. Because of the way the court is organised, it's even possible for the British judge in the EU court of justice to be the “rapporteur” for a major Schengen or euro case – but I don't know if it has ever happened.)