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1- Never, ever use pine or pine-oil. It will permanently stain the wodd white; and unevenly at that.
2- I tend to use Mister Clean or Lysol Brand floor cleaners...VERY well diluted in warm water. Never more than a quarter cup (230 to 250ml) in a standard 2-gallon bucket that is half-way filled to about 1 gallon(3.78L).
3- One can not use anything ammonia-based when one has a cat(s)...to them it means "pee here". Unless it is VERY well rinsed with lemon or vinegar. Then they won't go near that space! LOL
4- The cloth or mop must be damp; well wrung-out but not dripping.
Beware of oil-soaps like Murphy's. These don't clean well, and tend to lock the soil in under a coating of waxy, gooey, greasy "polish". Maybe ok to use AFTER REALLY CLEANING the floor, but over time this will leave the place dirtier than you started. Like Mop-n-glow. Scary bad, IMHO!
The tools one uses also makes a difference. Beware of sponge mops and Switfer and Swifter-like things. They dont really clean well. The only thing Swifter is good for IMHO is to keep you going back to buy more refills and products (NOT GREEN, and not a good cleaner, overall).
Try a good old fashioned string mop, or what I have found to work best is a cloth-strip mop. Mine is by O'cedar. With the right bucket (it's 'Bouquet', dear!) and wringer (see picture; insert and twist!) one's hands don't have to touch the water. [But then again if one is a princess to the degree that one doesn't want to get one's hands wet, one is not really cleaning well!!! LOL ) The whole mop-head pops off to be washed in a washing machine.
As always, start in an inconspicuous corner and see what happens.
PS vinegar is for rinsing and for ceramic tiles. I don't think it is best for wood floors.
My understanding is that you have real wood floors, but their top-coating (in that they were added over plywood sub-floors later) and that they were pre-finished right out of the box, is that their stain and finihh may not be as substantial as one is used to in homes that came with 'real' hardwood floors. BE CAREFUL!
