Eurail Global Pass. What the difference between budget, economy, and premium?

Probably something like that, but that depends on the railway company. For which one or for which country is this?

Usually the trains have just 1st Class (more service,leg room, more comfy seats)
and 2nd Class (the “Standard”)

Few trains have different travelcategory´s like for example the Railjets of ÖBB/Ceske Drahy they have ontop of 1st Class the Business Class :)

To assist you better i would ask to precise your question. as @rvdborgt  asks too.

And in some countries-to make it look more modern/hip, they have simply changed these old words 2nd/1st to some newspeak.

Note that in fact in many countries a fair lot of-though mostly only local/regional-trains only have 2nd cl. But most pass-newbees and USAers only tend to use those long INternat., trainrides that will have 2nd and 1st. Actually in most trains 2nd cl is far better in comfort as eco seats in planes.

Untill not that long ago-EUrail passes=for those who came from outside of EUR, were only sold as 1st for those a little adult over the junior age threshold of 26. That was a very old-time history to please the USAers who were brainwashed to think it was too uncomfortable to travel in 2nd. But maybe you got stories told from those old-time folks. So just to make sure: this is no more.