I'm not sure I know the definition of "reward night". My best guess of the intent of the badge is to represent how many nights are spent in a hotel per year, so "reward night" is short for "a night spent at a hotel that is eligible for reward". I think that is the hotel equivalent to how many miles are flown (which is the "Lifetime Airline Miles" badge).

I stay 365 nights per year in hotels (Marriott, specifically) and "earn" 365 nights + various bonus nights. According to my Marriott account, last year I earned 467 total nights (but at least 100 of those came from the annual double nights promotion and another promotion). I think some people might like the nights badge to represent nights credited (which can exceed 365) but for me personally, I think the badge would be best if it represented actual nights stayed in hotels.

(I don't know much about Hyatt but for Marriott Bonvoy, a lot of credit card partnerships provide quite a lot of night credits, and so it is theoretically possible to be credited for 100 "nights" despite having spent less than 25 actual nights in hotels. For that reason, I think the credited nights number is mostly useless, and actual nights stayed is much more representative of how a person uses hotels.)