I finally had the chance to read over the paperwork with Baxter's history, or what little there is of it.
Seems "Baxter" was a name assigned to him at PAWS, which explains why he occasionally seems to take a moment to respond to it.
He was brought in as a stray to ACCT, the city shelter, on 10/7. I am taking a guess that he was found with the other dog that PAWS said was his brother, because of the names the two dogs have. PAWS told me that Baxter's brother was named Pinky. Well, Baxter's intake paperwork at ACCT calls him "Brain". As in, "Pinky and the Brain". I am extrapolating some details, but my guess is that PAWS left Pinky's name the same, but changed Baxter's name, when they pulled the two of them out of the city shelter.
So I don't exactly know WHAT name my little guy grew up with. But he recognizes Baxter well enough that I'll keep that name. Not only has he been stressed enough already in the past 22 days, without having to learn a new name, I now THINK of him as "Baxter". So Baxter he will remain.