Your photos were not very clear, somewhat blurry (and no, it isn't my vision

) But from what I can tell, the floors appear to be in ok condition. More than likely, you will not be able to tell exactly what kind of finish is on them. Could be a natural or synthetic varnish, lacquer, shellac, wax, an early polyurethane, or a combination. If you do not want to have them resanded and refinished, the safest thing to do is to wax them with a high quality floor wax. I like Trewax. If you attempt to screen and recoat with a urethane finish, you risk having that finish peel off from contaminents or wax that may be present that you are unaware of. As I pro, I would not agree to recoat. I would only resand and refinish or clean and wax.