I think the full name is/was the PlayStation-X, but the X got dropped off somewhere along the line and only hangs around in the abbreviated PSX form. I'm not sure exactly where the "-X" stopped being used, but it might have been during the time when the system went from being the intended Super Nintendo CD-Rom add-on it was planned as to the stand-alone machine it eventually turned into.
I saw this last night and I first didn't know what it was. Given how much of a waste American Idol is, it was only a matter of time before somebody lampooned it. This is quite promising....as long as it doesn't launch any William Hungs.