Rimclick/Side Stick/Cross Stick are all gonna sound pretty much the same for practical purposes. Most drummers would acknowledge a difference between these and striking -just- the rim with a stick as the snare head contributes to the overall sound.
Snares can be fairly complex, depending on where you strike it (center to edge), how hard and which end of the stick you use. Mallets, hot rods, brushes, hands, etc. all expand the palette.
jackson white's post in Addictive Drums: Help with "minimalist drum map" was marked as the answer