Do presets sound different in Stage vs the other versions?

All the factory presets will be sonically the same across all three versions. You just have reduced access to change the sounds. 

With Stage you are highly limited in terms of how much of the user presets you can hear in the fxp section of these forums.

In Stage you won't even hear certain Pro users adjustments to the elements that aren't within Stage.

The Pro users can alter certain sound parameters that Standard users do not have access to, however Standard users can at least listen to/play with the efforts of the users with the Pro version, even if they just cannot adjust those settings for themselves.

Correct.

Stage (or Play as it was years ago) is essentially all factory presets hard baked using the same engine that the entire PTQ lineup uses.  I bought Play version 3 (which should tell you how long ago I did that), and then within a year or two upgraded to Standard and after another few years upgraded to Pro.  I got value out of all three.  Stage/Play worked excellently as I got started, and all of their presets were great: it just worked/works.  Standard was a great value upgrade as I needed it for sharing recordings as Stage was too limited for anything other than my practice needs.  Pro became needed when I wanted to recreate a lot of rare, antique pianos that I'd liked and the specificity of the sound control quality was necessary, and I wanted to tweak presets downloaded from the FXP corner as well in ways that Standard doesn't allow.  Essentially Pro gives you the note edit for nearly everything, while Standard only allows you to change the average value or baseline of a given note edit curve.

So, yes, I agree that wording is ambiguous in the documentation, but from my own experience, your upgrade plan sounds very solid.  I'm quite sure you'll be pleased, and of course, the trial/demo versions were extremely helpful on so many fronts for me.