With regards to the Magic Initiate feat (PHB pg. 168): It seems obvious that the 1st-level spell is treated as a special case with its own slot (this also seems to be the consensus from other questions on this Stack), but what about the cantrips? There's no unusual limitation on their casting, so they appear to be used just like ordinary cantrips. So, do they count against the number of cantrips a class can learn?
For example: under the "Spellcasting" section of "Creating a Bard" on page 52 of the PHB:
You know two cantrips of your choice from the bard spell list. You learn additional bard cantrips of your choice at higher levels, as shown in the Cantrips Known column of the Bard table.
In that case, if I'm creating a first-level human-variant bard, and I choose the Magic Initiate Feat, could I start with four bard cantrips instead of two, or would I still be limited to two bard cantrips at first level?
If the cantrips granted by Magic Initiate count towards my "cantrips known" limit, can the limit be circumvented by using Magic Initiate to get cantrips which aren't on the bard spell list?