Participles - can we call them PRESENT or PAST?

In modern grammar the progressive (‘ing’) form and the ‘en’ form of a verb are called participles. To distinguish between the two, they are termed as Active Progressive Participle (APP) and Passive Perfective Participle (PPP) respectively rather than PRESENT Participle and PAST Participle. The APP is called so because the ‘ing’ form of a verb appears in the active voice and in the progressive construction and in the participle (without a tense) form. The PPP is called so because the ‘en’ form of the verb appears in the passive voice and in the perfective construction and in the participle (without a tense) form. As such, can we assign the nature of PRESENT or PAST to them? They can, after all, be used in sentences of both PRESENT and PAST tenses. For example see the following sentences:

I am writing (Present). (‘writing’, the so-called PRESENT PARTICIPLE, appearing as PRESENT)
I was writing (Past). (‘writing’, the so-called PRESENT PARTICIPLE, appearing as PAST)

I have written (Present). (‘written’, the so-called PAST PARTICIPLE, appearing as PRESENT)
I had written (Past). (‘written’, the so-called PAST PARTICIPLE, appearing as PAST)

Is it not the tense auxiliary that makes the participle PRESENT or PAST?

Participles have no tense. Alone, they are non-finite.

Yes, indeed. Then it isn’t logical to call them Present or Past participle, is it?

It is fine: it reminds us of their origins and of the sorts of verb forms they are found in.