What’s the best cleaner for real wood floor with polyurethane finish?

I dunno why this is so hard but after years of living here I still haven’t found anything great.
At the home center every time I’ve asked the staff seem bewildered by the idea that the floors are real wood and not some engineered material…. Then they try to convince you it’s not real wood, so I show them pictures of the flooring being installed and video of me doing the finishing. Then they do this look 😐.

The one I used most often from Conan or viva (can’t remember) cleans the floor well enough but if you don’t mop it up really fast it leaves permanent etching in the finish. I threw it down once and worked the mop around the area and it has permanent splash marks etched in now.

And if there’s anything specific to canine-bio-war-crimes that might be helpful as well. I’m worried eventually my house is gonna stink before my dog gets his life figured out 😑.