I have a trackball that looks almost identical to yours, except instead of a "H" mine says "K M". Bought it from China few years ago, but never used it.
Just plugged it into my XP machine, and it works via PS2 only (as a standard PS2 mouse). I have a passive PS2 to USB adapter but I can't get XP to recognise the trackball that way (although the LED light inside the ball does come on). So, just same as your experience.
Exactly the same experience with my Win7 machine. It only works via PS2 unless there is some arcane thing we don't know about yet.