I was mainly using this rotary with my fiber laser. It came with two sets of jaws (internal and external) and could grab most anything I threw at it.
More recently, I picked up this low-profile rotary specifically for engraving Contigo travel mugs. It has reversible jaws that bolt on with three small screws each, and it has enough capacity to grab the rim of my mugs, but I might print some bespoke jaws for the mugs.
I got the low-profile rotary, a set of tower extensions, and a 300x300mm lens so I could do single-pass full-bleed art on the compound-curve Contigo mugs, so if you’re just doing “normal” mug engraving, you wouldn’t have as many constraints.