How to protect a painted floor

I honestly do not think painting a floor would be a good idea. A solid color paint or stain will likely chip and wear unevenly. As in wherever your high traffic areas are will look awful in no time at all.

If you are to paint it you will need to sand the gloss of the previous coating. This glass will prevent any other paints or stains from correctly adhering to the surface. The same exact rule applies to painting a glossy wall or any other surface.

Try some very fine steel wool, #0000 should do it. Confine it to the paint spots only, as best you can. You may try using small amounts of the spirit you tried with that. If the wool dulls the shine of the floor finish test an area first and apply wax over the treated area when done to restore the luster. You could also try automotive scratch removing compound. Use a little bit on a rag and buff the paint out.