Saul Goodman character made his appearance in Breaking Bad's Episode 8 of Season 2 named "Better Call Saul". In the "Insider Podcast" of the episode, Vince Gilligan explains that he came up with the name "Saul Goode" (it's all good) one day as he was showering and that it was later changed into "Saul Goodman" (it's all good man) after the suggestion of one of the other writers, J. Roberts.
Here's the Podcast where Gilligan tells the story (circa 1:25 into the clip):
So, to answer your question, yes, the joke on the name was completely intentional and, while in the same episode of BB Saul explains to Walter that he chose the name in order to sound "jewish":
“My real name's McGill. The Jew thing I just do for the homeboys. They all want a pipe-hitting member of the tribe, so to speak.”
in Gilligan's intention the play on words was meant to appeal to the criminal clients that Saul's law firm is targeting.
As you pointed out in your question, the "fictional" genesis of the name might be explained in Episode 4 Season 1 of Better Call Saul ("Hero") where a younger James McGill (not yet Esq. and known at the time as "Slippin Jimmy"), introduces himself (for the first time?) as “Saul” ... “S’all good, man.”