The 360J is .38spl only, mine is part of the first run from 2008, I bought it from Bud's. In late 2008, maybe early 2009 Bud's was selling some .357mag 360's that looked like the 'J' model but they were not 360J's nor were they M&P's. They were selling for about $100 more than the 360J. As far as I know that was the only time they were available, haven't seen them since but there's been several runs of 360Js in that time.
I looked at the gun in the auction, that is very similar to the one I'm talking about but I can't see the date on the empty casing envelope to be sure and the grips look later than '08-'09 but I could be wrong.
As far as what gun someone is referring to when they say '360', who knows? People say all kinds of things that mean something else. I know guys that still refer to every 1911 as a 'Colt .45'. You have to ask them to be more specific, I guess.