Wood – Do chips in a stained floor reveal differently-colored, unstained wood

This is an educated guess only because I can't really see or feel what you are describing. This may be old floor wax that has had something damp left sitting on it. Often in older homes folks would use old fashion floor waxes that will discolor or turn white with age, water, friction or heat. You may first try using a little mineral spirits or a mild cleaning agent with a polishing compound, such as Bar Keepers Friend with a soft cloth on a small spot to see if it will remove the white residue. If in fact this floor has been treated with a wax product, the entire floor would have to be stripped of wax, perhaps buff sanded and restained, then protected with a few coats of Urethane. Don't attempt to apply urethane until you have determined what is causing the white waxy residue. Putting any coating such as urethane over a waxed floor will bubble, peel, and create a mess to clean up. another test you could do would be to gently scrape a very small section of the white stain with a putty knife. If the material comes off easily it may be wax. it may be old varnish, but that would not feel waxy, but not out of the question. Let us know what you find, and we can go from there. Good Luck

I've seen kits of cleaner/refinishers - I have no idea how good they are, but for $20-$30 they might be worth trying before bringing out the heavy machinery.

A quick search came up with this - I'm sure your local big-box will have something similar.