I believe originally it was simply called the "Playstation", this was the gray model. Then in the summer of 2000 the PSone was released, optimized for easier portability along with a newer graphical interface. The PSX on the other hand is a common misconception; this was a Japanese exclusive Playstation and DVR hybrid.
So in short:
Playstation for original release.
PSone for second generation Playstation.
PSX for the DVR hybrid.