How do I keep paint from chipping off the jaws area of my old vise?

You didn't ruin anything, but yeah, those machined slide surfaces should stay bare. Paint on there will just keep flaking off into whatever you're clamping, and it gums up the slide anyway. No point letting it wear in, that's just going to shed paint chips for months. The acetone and rag thing is a start, but honestly on fresh enamel it'll probably take some scrubbing. I'd use acetone on a shop rag and work it in with a scrub pad or even a razor blade held flat, depending how thick it went on. If you built up a few coats it might take a couple rounds of solvent and scraping. Mineral spirits works too, just slower. Once it's down to bare metal, hit the slide bar with something like a scotchbrite pad to knock off any residue and give it a nice smooth surface again.

For keeping it moving, oil is fine but don't drown it. A light coat of way oil or even 3-in-1 on the slide bar and in the screw threads, then work the vise open and closed a bunch of times to distribute it. Wipe off the excess so it doesn't attract grinding dust and swarf, wich is what actually galls slides up. Some guys run dry lube or just wax on the ways since it doesn't collect grit the same way, but plain oil works if you remember to reapply now and then.

One thing to consider for next time on a restoration like this, mask the slide faces and the screw before painting. Or paint first, then dress the contact surfaces with a file and stone after. Either way you end up with bare metal where metal meets metal, which is how these were meant to run anyway. If the paint edge at the boundary of the slide looks ragged after cleanup, a little touchup with a fine brush along the line cleans it right up. No need to redo the whole job.

Same thing happened to me when I cleaned up an old bench vise a few years back. I sprayed right over the slide bar without thinking, then spent an evening picking the enamel off with a single edge razor held flat like that. Took maybe twenty minutes and the surface underneath was still fine, no scratches. I agree with letting the bare metal stay bare after that. I just ran a fine stone over it lightly and it slides like it always did. The touchup brush tip on the paint line is a good one too, that ragged edge bugged me until I did the same thing.

I totally get the anxiety about damaging the slide when you try to clean off thick enamel. I had a similar problem with a clamp guide rail once, and while the chemicals get most of it off, I found that using a small bit of fine jeweler's rouge over the stripped spot was the trick that really made it smooth again. It took forever, but it made the contact area glass smooth, not just scratched. I always feel that leaving the metal bare is the right call for moving parts, even if it looks rough right after stripping.

Update: went with the razor blade trick, held flat like the one guy suggested, and it took the bulk of the enamel off the slide bar in like half an hour. Looks bare again. But there's still a thin shiny film down in the machining marks that the blade just skates over. Is that thin leftover going to gum things up, or will oil and use eventually flush it out?