They've never mentioned the name. I guess it's just "The City". And yes, it does seem to be located where Puerto Princesca is, your friends must have been around on Bnet during the darkness.

The name of the Last City is "The City", because it is the last safe city on earth, it needs no distinction, because there is only one city left. Characters and members of the Bungie team refer to it as such. It is [i]speculated[/i] that it's location is where the city of Puerto Princesa stood before the calamity that destroyed it and all other places on the Earth. After the Traveler's arrival, the last survivors went there (most likely because they wanted to see why a giant orb was floating in the sky) and, as more and more came to what was now the only real landmark left to go by, they built the City underneath it because of the Traveler's magic and protective nature.