It depends on the definition of "tense".
In a narrow technical sense, linguists define "tense" as an indication of
when the action described by the verb occurs. (It's slightly more complicated than that, but that's close enough for the moment.) Since the present progressive really does take place during the present, it's just the present tense. In other words, tenses are just windows of time-- generally only past, present and future, although actually some languages have multiple past (or future) tenses such that there might be a "since yesterday" tense versus a "distant past" tense, etc. -- still, those refers to windows of time.
But then there is
aspect which further adds information about
how the verb interacts with time, or how the events are carried out relative to the tense. The progressive is an "ongoing" aspect, versus for example the perfective which is an "already completed" aspect. So "I have eaten" is
present perfective, because it's a present tense but perfective aspect (the eaten finished before now, with the idea of "present relevance" explaining why this is present tense). Compare also the past perfective "I had eaten", referring to some time in the past, before which the eating had already been done.
If you want to read more you can find information on Wikipedia, etc., but there are two very good books to refer to, simply named
Tense and
Aspect, written by Bernard Comrie. (They're popular and well known so it won't be hard to find them at any academic library.)
On the other hand, more loosely, many non-linguists, especially language teachers (for practical purposes, and sometimes casually a few linguists, will define "tense" as simply any sort of verb form. Usually it's limited to actual prefixes or suffixes on the verb, so you might hear someone say, for example, "has" is the "third-person tense" of "have" (instead of "person agreement"), or the "subjunctive tense" (which is a mood). That's sort of a bizarre use of the word and confusing, but makes sense when you know they just mean "verb form".
It's rarer, but still happens sometimes, that people will describe a complex form like "be + V-ing" as a "tense", although more common as a "complex tense" (or similar wording). More technically we'd call that a periphrastic (multi-word) verb form.
In short, language teachers (and others) just use the shortcut term "tense" to refer to "ways we say verbs". It's not very precise or clear, but when your goal is just explaining different verb suffixes in a textbook I guess it works.
Or is it a Oxford comma situation, where it just depends on who's rules you accept at the start?
Yes, sort of. But there isn't much uncertainty about the data (at least for linguists). Instead, in this case, it's about which
labels you use for what. The actual things you're describing don't change. (The oxford comma is a separate issue, related to prescriptivism, telling people how to write, rather than describing how people actually write. Although the label itself just describes that usage I suppose.)
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As for the broader question of "how many" tenses English has, it of course depends on the definition.
Narrowly, English clearly has three tenses (plus various aspects, etc.).
More broadly, there are dozens of possible verb forms if you include multi-word forms: for example, "used to", or "gonna", etc.
