Unrestricted depends on the game, usually it's completing something without any restrictions on glitches, or a certain in-game item, or other exploits. Make sure to check the game's rules and category rules.
NMS, as far as I've ever seen, just means 'No Major Skips' and means one or many major glitches, exploits, or indeed skips, have been banned from that category but might be allowed elsewhere. Again, check the category and game rules for exactly what the category is forbidding.
If you do want to know exactly what's different about these categories for a specific game you can always try asking in the community, or a moderator directly failing that.
Lots of games have extra categories involving getting all collectibles, or 100% categories involving that kind of thing.
Assassin's Creed's series is a terrifying example of that https://www.speedrun.com/series/assassins_creed
D.W.W.R - 'Do What We Rehearsed'
-xem92 (or possibly somewhere else in the Ratchet community) this is just where I've heard it, and honestly for complicated speedruns it's great and helps you stay level headed and avoid panicky mistakes.
they've seen more instances of duplicate games getting added.
That's a shame, I hadn't previously noticed duplicates (except I think on one or two occasions where a series moderator added something which already existed elsewhere). Part of me hopes there's some kind of explanation to duplicates cropping up other than carelessness, and that it's not for us to see, but hey.
Obviously you're right consistency is very important; having the two moderators come to a decision is ideal, even if by default I personally would hope to see the original board given the space to work in (especially if there's an active community around it, although this could become true of a new board if the mistake somehow went unnoticed for a while).