[RPG] Do cantrips granted by Magic Initiate count towards “Cantrips Known”

This looks to be entirely up to DM discretion.

There are two equally valid positions on this and the rules do not refute either one.

The first is that this is a completely separate spell, and slot, and that the two have no overlap with your casting class. The spell you learned and the slot you obtain are kept separate from the class spells if you have levels in that class.

The second position is that it is a class spell, from your class list and you can retrain it as you would when you take a new level in that class.

The second position is mildly problematic in that you can continue to obtain higher level spells with the retraining, and it may lead to this MC being simply the obtaining of another 1st level spell slot, rather than feeling like a mini MC.

The first position is problematic in that it introducing additional bookkeeping and may be perceived to be overly harsh and pedantic.

In my games, I think I would allow the MC feat to simply be an additional first level spell slot and allow the spell to be retrained if it's from a class that allows that and you take levels in that class. I would probably not make them remember which spell goes with their MC beyond that the extra slot must be a spell from that class. This may be too lenient for some games though.

Your understanding is correct

The Magic Initiate feat, as you've correctly noted, lets you learn two Cantrips, and Cantrips scale with your character level, not your caster level. This means that your Initiate-Rogue can get the full, scaling power of their Cantrips for the rest of their life! You are also correct on how to calculate your spell attack bonus, using the relevant Ability modifier and your proficiency bonus.