The original Play Station (commonly written with the space) was an SNES CD unit designed by Ken Kutaragi himself (who also designed the SNES sound chip), which was the source of a major coldwar between Nintendo and Sony, as well as a lot of rumors and misinformation.
When Sony finally gave up and went their own way, they announced development on what they were calling the "PlayStation-X", which was widely believed to be keeping the original PlayStation's name as an f-you to Nintendo. I don't think it ever had an "official abbreviation", but the videogame media took to calling it the "PSX". The final version was just called the "PlayStation". But the "PSX" nickname had already caught on, and everyone was calling it that, including people at various divisions of Sony.
Around the time when the PSX became the undisputed winner of the 32/64-bit era, Kutaragi decided that he was tired of being reminded that the PSX was his "second try" at the console industry every time he saw the name, and wanted to create the image of a flawless record, so he insisted that the PSX was the "original PlayStation" and demanded that people stop using the "PSX" nickname for it. Then when they were redesigning the PlayStation, he decided to invent and push the "PSone" nickname, hoping it would replace it. This is why the redesigned PS2 wasn't officially renamed the "PStwo". There was no need for a new name.
When another division of Sony made a Tivo-like device with a built-in PS2, they named it the "PSX", saying that if Kutaragi didn't want the name, they would take it, and this apparently got him really ticked off.