Deore RD M592 with old school shimano trigger shifter I have an old school Devinci Coyote (2002) whose Alivio rear derailleur is having trouble shifting properly. I was wondering if I could replace it with a new Deore M592 without having to change the trigger shifters (tight on the budget).

Searching on the internet seems to hint that shimano has been sticking with the 2:1 pull ratio for a long time and this could possibly be a direct drop in without much adjustments. But I would love to hear your opinion on the derailleur and the compatibility before I pull the trigger on the purchase.
Yes, it will work fine. RD-M592 is designated as a 9-speed rear derailleur. All nominally 7, 8, and 9-speed Shimano mountain RDs have the same actuation ratio.

In theory putting an 8-speed chain through the cage can cause issues, as that's the area that could be sized to want a narrower 9-speed chain, but in practice it never seems to. RD-M592 in particular practically exists to be a nice but not too expensive legacy replacement rear derailleur for 8 and 9 speed mountain bikes, and I've never seen it have issues being used in this capacity.