I want to replace the 42t chainring on my old Sirrus (Suntour Blaze 7-speed) with a 38t or 39t. I'm finding lots of chainrings described as 10-speed compatible, some as 9-speed, some as 8-speed, some as 7-9 speed, etc. And some that don't specify at all.

Can anyone tell me what actually makes a chainring work with one configuration and not another? And should I follow exactly what the descriptions say, or would, for example, anything but 10sp work with my setup?

(I especially don't get how one could be labeled just 8-speed and another could say it works with 7, 8, or 9.)

The CW are slightly thicker for older setups, but 7-8-9 are relatively
compatible as are 9-10, such that you would not notice much.
A 10spd on a 7 will still work. A non specified is likely to be 7-8 spd.