I use Bona for hardwood but it has left my matte very dull. It is so frustrating but our flooring guy said no water on hw. I just don't think water can hurt. There has to be a lot of water in any spray on cleaner.

I use a dash of vinegar and 2 drops of dish detergent in a bucket of warm water. I use a microfibre type string mop and wring it out really well so it's barely damp. The floor looks like new. I prefer the KIS principle when it comes to cleaning and prefer not to use a lot of unnecessary chemicals or scented products.

What are the white marks from?

I used Method cleaner with a dust mop - a lot of people recommended it and said that it's safe. Also it smells like almonds. Can't complain about that.

Here is a link that might be useful: Method Cleaner

I use my Haan steam mop on my pine floors and love it. No chemicals, easy as pie, sterilizes, dries very quickly, and Haan says it's fine to use on hardwood floors (although pine is a soft wood). Key, though, is to keep it moving, never let it sit idle on a wood floor.

I use Bona on the engineered floor with poly finish in one of the bedrooms. Water and vinegar for the wood floors with tung oil finish in the rest of the house.

A very small amount of vinegar in warm water for my engineered wood floor.
1/2 cup white vinegar into 1 gallon warm water.
No "swab the deck" type cotton mop. Sponge mop or a rag wrung out really well, no puddles while you're cleaning.
I use a towel to wipe dry afterwards.

Here is a link that might be useful: Using Vinegar for Cleaning Hardwood Floors

The Bona system works so good, you get the mop and the spray bottle of solution, spray it on, wipe it off, done.