Do crank arm pullers vary in sizes or there is just one size?
The main distinction is whether the puller is for square taper (and Powerspline) versus Octalink/ISIS. Among square taper crank pullers, there are some older ones made with nutted spindles in mind that don't play nice with the now more common bolt-style ones because their tips are too narrow and can jam into the bolt hole and damage the threads. A couple companies now have good all-in-one tools with switchable tips that work on both square taper and Octalink/ISIS. 

There is no road versus mountain distinction with crank pullers.

One size of extractor thread has been used on the vast majority of cranks for a long time, M22x1. There are several older sizes on cranks made by T/A, Stronglight, and possibly some more obscure ones, all of which need their own extractors. There's also some more contemporary brand/model specific designs such as the Profile ones and Cannondale SI.

Within BMX, there's now about a million model-specific crank removal tools, and a whole other discussion of cross-compatibility.