Not in general, no.

All of Shimano's 8-speed and 9-speed derailleurs use the same actuation ratio, which is to say, the amount of lateral movement that the derailleur does for a given amount of cable pull. That's why you were able to use the "9-speed" derailleur on the 8-speed drivetrain: Shimano 8-speed and Shimano 9-speed rear derailleurs are basically the same thing.

But when Shimano moved to 10-speed, they changed the actuation ratio of their mountain derailleurs. So a 10-speed Shimano MTB derailleur not index properly in an otherwise 8-speed or 9-speed drivetrain.

Where this gets really complicated is with the road parts.
When Shimano first introduced their 10-speed road stuff, they used the same actuation ratio as the 8/9-speed stuff. So the early 10-speed road derailleurs were largely interchangeable with 8/9-speed derailleurs.
When they introduced their 11-speed stuff, they changed the actuation ratio, making the 11-speed road derailleurs incompatible with their sub-11-speed drivetrains.
BUT.
In 2015, Shimano introduced the 10-speed Tiagra 4700 groupset. This groupset used the same cassette specification as previous 10-speed groups, but the rear derailleur uses the 11-speed road actuation ratio, with the shifter pulling however much cable per click to make this combination work. This is how Shimano does 10-speed road in general now; the newer GRX400 gravel groupset works the same way.
One of my friends recently broke the 11-speed R7000 rear derailleur on his gravel bike, and replaced it with a "10-speed" RX400, and it works great in the 11-speed drivetrain.

You're mostly right, but see above. Shimano's RD-RX400 rear derailleur is nominally a 10-speed road derailleur with a clutch, but it won't work in an 8/9-speed drivetrain because it uses the post-2015 Shimano 10-speed road spec.