Many different terminologies have been in use over the years and there is still a lot of variation.
(Present) Perfect
"I have done" can be called the perfect, the present perfect, or the present perfective depending which grammar you look in (though I believe the term "present perfect" has come to dominate, and this area of agreement is to be welcomed).
Quirk et al. regard the perfect/non-perfect distinction as one of aspect, not tense, and use the term "perfective".
Huddleston & Pullum regard the distinction as one of secondary tense and use the term "perfect", so to that extent they support the traditional notion of a compound tense (analytic tense) rather than simply recognising simple tenses (synthetic tenses). Many older grammars regard the various perfect and progressive constructions as "compound tenses"; many newer grammars say that English has only two tenses, present and past, and that "I have done" combines the present tense with the perfect(ive) aspect.
For H&P, the present perfect combines the use of the present tense in the primary tense system with the use of the perfect tense in the secondary tense system. (By contrast, Quirk et al. and H&P are in agreement that "I am doing" represents the progressive aspect, whereas, again, older or more traditional grammars might describe it as a "compound tense" called the "present progressive tense" or something similar.)
The terminological confusion also encompasses Latin. Originally what we now call the perfect or present perfect was called the past perfect (and indeed, British learners of French are still often told that the perfect tense is a "past" tense", and the French themselves call it the passé composé, compound past). In 1889, an article in the journal Academy[1] noted: "The form ‘scripsi’, the traditional ‘past-perfect’, was now called ‘present perfect’; ‘scripseram’ was called past-perfect".
Although Quirk et al. use "perfective" to mean "perfect", H&P use "perfective" with a very different meaning (the simple past is "perfective", according to H&P, where it describes a completeted event in the past). So what appears at first to be a minor difference in terms has extra significance.
Past Perfect
Similarly, the past perfect is also known (but today less commonly, although in study of foreign languages the older term is still often used for the equivalent forms) as the pluperfect.
Perfect Infinitive
Because Latin has a distinct morphological form called the perfect infinitive, it was natural that the equivalent periphrastic or compound form in English - "to have done" - became known as the perfect infinitive as well.
I don't find this inconsistent with the terms "present perfect" and "past perfect", because in all three terms, the term "perfect" represents the fact that the perfective aspect (Quirk) or the perfect secondary tense (H&P) is being employed, i.e. the fact the auxiliary "have" is being used followed by (what is often misleadingly called) a past participle.
It has been suggested in the comments that "infinitive perfect" would better parallel "present perfect". But "infinitive" is (according to some, but not all!) the name of a mood (though others consider the whole concept of mood largely irrelevant to modern English anyway: for example, H&P consider the term "indicative" essentially an irrelevant archaism and only mention it twice, once in a footnote; they do use the term "mood" but primarily to denote the analytic system of mood represented by modal auxiliaries). If the infinitive is a mood or similar to a mood then the term "infinitive" in "perfect infinitive" isn't comparable to the term "present" in "present perfect" and doesn't need to come in the same place in the phrase. We probably speak more often of the "present indicative" than of the "indicative present", though both terms are in use.
Quirk et al. use the term "infinitive perfective" to describe what is often called the "perfect infinitive". So they do in fact put the term "perfective" at the end of the expression here too. (Note: just as the term "infinitive" does not refer solely to "to"-infinitives, Quirk et al.'s term "infinitive perfective" does not refer solely to "to have done" but also to the words "have done" in a phrase such as "may have done".)
As far as I know, H&P do not give a specific name to the form "to have done". They regard it as a nonfinite clause involving a "to-infinitival" combined with a verb to which the perfect secondary tense has been applied.
Past and Present Participles
Traditionally "done" (without "having") is called the past participle. This is hugely illogical, because it is used as part of the present perfect ("I have done") and future perfect, not just the past perfect. So the term "perfect participle" would make more sense (and I believe it has occasionally been used, with the present participle renamed "progressive participle"), only it would be confusing because of the more common meaning of "perfect participle" (as below). It would also not represent the fact that the identical participle is used in passives ("it was done"). Others have suggested the term "passive participle" (with the present participle renamed "active participle"), but this wouldn't represent the fact that it is used in perfects, and the parallel term "active participle" has the problem that is also used as part of passive expressions ("it is being done").
Overall, the term "past participle" has survived, but Quirk et al. replace it with ther term "-ed participle". (Of course, it doesn't always end in "-ed"; sometimes it ends in "-en", for example.) They similarly replace the term "present participle" with "-ing participle", which is understandable because the term "present participle" is a particularly illogical way to describe a form that occurs in expressions such as "I will be doing", "I was doing", etc.
H&P use the traditional term "past participle"- although they are not happy with the term "present participle" either, emphasising that "it is not a tensed form of the verb". So they replace the term "present participle" with "gerund-participle".
At this point it is important to note that neither Quirk et al. nor H&P (the two major comprehensive English grammars) regard the distinction between "present participle" and "gerund" as a useful or valid one. So H&P use the term "gerund-participle" because "there is no justification for making any inflectional distinction... we call this form gerund-participle to reflect the fact that it covers the ground of both gerunds and present participles in other languages". Quirk et al. say "We do not find it useful to distinguish a gerund from a participle, but terminologically class all the -ing forms [that are traditionally held to be gerunds] as participles... [The] lack of correspondence between the English gerund and the traditional use of the term can be held as a further reason for rejecting the term gerund in English."
Perfect Participle and Perfect Gerund
I'm not sure how widely used these terms are outside the field of EFL/ESL, or whether they are traditional. In any case, based on their lack of support for the distinction between gerunds and participles, it follows that neither Quirk et al. nor H&P would use the term "perfect gerund" or regard it as a distinct form from the perfect participle.
H&P regard "having done" as a "gerund-participial" used with the perfect tense. (A gerund-participial is a gerund-participle used as part of a gerund-participial clause. So "having" on its own can also be a participial.) They don't seem to regard it as requiring a specific name.
Similarly, Quirk et al. regard "having done" as a participle clause (they don't use the term "participial") used with the perfective aspect.
Sources:
- Quirk, R., Greenbaum, S., Leech, G., Svartvik, J. (1985) A Comprehensive Grammar of the English Language, Longman.
- Huddleston, R. and Pullum, G. K. (2002) The Cambridge Grammar of the English Language, Cambridge University Press.
[1] Cited in the OED Online under 'past perfect', a sub-entry of 'past'; https://oed.com/view/Entry/138567?redirectedFrom=past+participle#eid31698049
