I think it may be just the searches I've been doing.
Hetts press apparently prints directly from the cylinders, and I've been looking at offset presses, which are credited to Rubel just after 1900. Invented 1901, and in production by 1905.
https://www.si.edu/es/object/nmah_882246
Multi color offset presses are basically just two or more of those strung together with a common feed and output. And in some ways hadn't changed much into the 1980's.
I've been reading some of his patents, and they're very interesting. The printing cylinders were a copper tube with a cast zinc surface that could be used for several other types of printing. The patent I read didn't specifically mention lithography, but an article about it did. That use may be in a different patent, and the surface may have been something other than zinc, as I can't picture that being a good material for water retention. But who knows? I may have to experiment, zinc plates are used for etching and zinc plated plates for corrosion resistance, and they're not expensive.
More interesting to me but not at all applicable to cards, is that it looks like different types of printing may have been possible on the same machine.