I use a cloth simply because I don't normally buy sponges. (Not for any environmental reason, I just don't). I don't rinse Murphy's, but I dilute it more than the instructions say. Instructions say "Rinse". Because your floor is as bad as it is, I would have wash water, rinse water, and a dry rag. And the full amount of soap. Once it's clean, and if the finish is still top-notch, you can just use a wet cloth for touch-ups, no soap required.
"Let me see you stripped down to the bone. Let me hear you speaking just for me."