Learn English – Tense when saying “This is the first time I”

This is a bit clearer if rather than "still relevant" you think of the perfect tenses as expressing a current (with respect to Reference time) state, not an event.

The English phrasal verb be born, however, designates an event: birth, which is the punctiliar fact of entry into a state of extra-maternal existence (I am being as pompous and mealy-mouthed as possible to avoid politico-theological controversy). An event is expressed using a simple form, not a perfect form.

There are circumstances in which a perfect construction with born is proper; for instance:

A son to the Earl has now been born, so his nephew Roderigo is no longer the heir apparent to the title.

Here a current state is spoken of: the existence of an heir presumptive. But if you want to locate the birth in time you are speaking of the event, not the consequent state, and you must employ the simple tense:

A son to the Earl was born yesterday, so his nephew Roderigo is no longer the heir apparent to the title.

In situation 1, which version you select (they're both grammatically correct) depends on the tense in which you're telling the story. If you're telling the story in the present tense ("So I'm walking along the street..."), you would use option A to match, while if you're telling it in the past tense ("So I was walking along the street..."), you'd use option B.

In situation 2, you'll generally want either A or C.

The essence of the grammar article was probably that in English, it's possible to discuss either past events or hypothetical future events in the present tense, as if you were placing yourself in the time and narrating from that perspective. In this case, you'd use the present-tense "first time I've been", but it's much more common to narrate this way when relating past events than when talking about future plans.