I can't speak of this with any certainty, but I believe that all three names are the same place.
Many Welsh place names were named in Welsh, and then given an Anglicised spelling. In the past the anglicised version became the accepted name, and the original drops out of use. Someone then comes along and tries to do a back-formation for the Welsh name, sowing confusion.

Things can get even more complex when you add in the Welsh mutations.
Carmarthen is (possibly) named from "Merlin's fort", Caer Myrddin" but due to a 'soft mutation' it becomes Caerfyrddin.
But the initial letter 'C' can also mutate in three separate ways, so although the town is Caerfyrddin, the expression for "To Carmarthen" is "i Gaerfyrddin" which is the 'soft mutation'
There are also the 'nasal mutation' Nghaerfyrddin, and the 'aspirate mutation' Chaerfyrddin.

For the same reason you'll see posters for the rugby "Cymru am byth" (Wales for ever). But when you cross the border you'll see roadsigns "Croeso i Gymru" (Welcome to Wales). Again the soft mutation has altered the initial letter.

So, a simple query, but a complex answer.