Dust cover up or down?

My experience is, it all depends.

With my Belt drive Music Hall MMF-5 I had to play a disc with the dust cover off.

With my direct Drive Technics SL 1210 GR it does not matter which way, closed, off, half way open, it's all the same. No effect on the sound.

soko, I don't agree with your interpretation.  There was a time in the history of audio per se as a hobby, let's say since the mid-60s,  when the dust cover was an expected part of a turntable, and many manufacturers provided a hinged attachment for convenience.  I never took that as an indication that the manufacturer was recommending anything.  It was just a feature that might seduce a buyer.  In many cases, where there is a hinged DC, the hinges are designed such that the cover can be totally disengaged from them to allow for their complete removal.  I wouldn't over-interpret that, either.

Off

Goes without saying. Off!

Down. To prevent dust settling in the record.

If you are in a fairly dust free environment, then, you could leave it up for convenience.

For me in industrial Singapore, I see benefits from keeping it down.