No experience with Ha-Ra but we recently invested in a Karcher brand steam mop as the majority of our house is tiles.

I love it and don't know why I didn't get one sooner. No more struggling with the bucket and dripping water onto the floor etc. our tiles are not high gloss but I can't image why you wouldn't use it on them or timber, it is very gentle but gives a really through and effective clean.

You can also add floor cleaner to the water in the machine but I have not found it necessary, just the steam itself is enough to get any stains off the floor.