In school grammar, "present perfect" is a tense. School grammars are basically tradition, so you can call it whatever you want as long as you agree with long-dead grammarians. Ditto for Turkish - I'm sure it has its own dead grammarians.

In modern grammar, "present perfect" is not "close to a tense" - it's a combination of present tense and perfect aspect.

We can say a little more; in traditional grammar, aspect is not a recognized category. Thus, while it is very clear that Latin has a system of three tenses, two aspects, and three moods (counting imperative), traditional grammar assigns it six "tenses":

    +---------+-----------+------------------+
    |  tense  |   aspect  | traditional name |
    +---------+-----------+------------------+
    | present | imperfect |     present      |
    +---------+-----------+------------------+
    |   past  | imperfect |    imperfect     |
    +---------+-----------+------------------+
    |  future | imperfect |     future       |
    +---------+-----------+------------------+
    | present |  perfect  |     perfect      |
    +---------+-----------+------------------+
    |   past  |  perfect  |    pluperfect    |
    +---------+-----------+------------------+
    |  future |  perfect  |  future perfect  |
    +---------+-----------+------------------+

This is the reason for calling perfect a "tense": it's traditional. But this model won't stand up to analysis. Interestingly, the Romans themselves do not seem to have used it; where we refer to "pluperfect tense", they referred to the "past perfect-er tense", identifying both tense and aspect (admittedly, both under the name "tense", or rather "time"). I don't know when the conceptual distinction was lost.

To nitpick: under modern analysis "future" is not a tense in English: the future verb "will" (or "shall") behaves much more like "can", "may", "must" and so on - they're collectively called modal verbs, i.e., in English future is a mood.

A tense expressed through a modal verb is still a tense.

Linguistics does draw the distinction between syntactic "tense" and semantic "time", but in that case English modal verbs wouldn't reflect "moods" at all, just "modality". They're all periphrastic. The same goes for perfect aspect, also periphrastic in English, though I don't know offhand how (or whether!) there is a terminological difference between syntactically-marked aspect and semantically-present aspect.

The same objection would theoretically apply to voice, where the English passive voice must be periphrastic too, but in that case everyone agrees that this is a distinction of voice and the difference between inflection (where grammatical meaning is expressed by changing the form of a single word) and periphrasis (where grammatical meaning is expressed by combining multiple words) isn't relevant. This is just an inconsistency in modern theory, which probably arose because voice isn't relevant to semantics at all.

Ignoring the modal auxiliaries, English would still have moods, subjunctive ("We demand that Robert be ejected from the book club") and irrealis ("If Robert were to be ejected from the book club, ..."), but neither of those is in a particularly robust state in the modern language.

Inflectional constructions express tense through morphology and periphrastic constructions express tense through syntax. Together they constitute expressions of grammatical tense.

> Why is "present perfect tense" closer to a tense, but "learned past tense" is closer to a mood?

Are you thinking of these as exclusive categories? Every finite verb has a tense and a mood. That's the point of having separate terms; these are independent dimensions of the verb.

Theoretically, there could also be a "reported present" verb form, except that this is semantically impossible: any event that has been reported to you must have happened before the report did, and the report must have happened before you started talking about it, so reported events are stuck in the past.

It's possible, though, to imagine someone making a statement about reported information in the future, in which case the event would take place before the report, but possibly after I describe how I'm imagining the future. Would anything interesting happen in Turkish for this kind of sentence?

-mis'li gecmis zaman or "inferential past tense" is "inferential mood" and "past tense".

Essentially old school people categorised tenses out of thin air; and modern linguists define tenses as "time reference",mood as "modality signalling" that is "relationship to the reality / truth" and aspect as "expression of how something extends over time". So aspect doesn't apply here.

So -mis'li gecmis zaman is a tense and a mood. Sometimes.

Sometimes it is something altogether different such as mirativity (https://en.wikipedia.org/wiki/Mirativity).

PS: I have to say, I'm surprised your handle in HN isn't @ssg :)