In most cases the pull direction is determined by the cable routing arrangement on the frame. Some frames do give you the option but that's not super common. Most new clamp-on MTB derailers are dual pull (they can be set up for top pull or bottom pull).
You also should match the front derailer to your chain type in terms of width/speeds. The difference in the cage width is real albeit subtle. One may not encounter issues from this, and it's possible to work around it by angling or bending the derailer, but if you want it to have a better chance of just working then get a 9-speed one to match the bike. (Bikepedia says your bike is 9 speed and came with a 9-speed Deore FD. The STX-RC derailer in question here is actually 8-speed, which I presume got stuck on as a less than ideal replacement at some point. If this is incorrect and the bike is actually 8-speed, then get an 8-speed FD).
Virtually all (maybe actually all) 9-speed and earlier mountain front derailers are triple. There are two ways of making the distinction. The good way is double checking on a source like techdocs.shimano.com or just googling the model code and finding the "Service Instructions" pdf for that derailer. My search history for example is littered with things like "fd-mc36 si". There's also the "Specifications" area on techdocs.shimano.com. The other way is recognizing how triple FDs for both mountain and road have a bigger, more dropped-down inner cage made to reach across and contact a chain that's sitting on a lower, smaller ring. If you look at some pictures you'll see.
You should also match whether your existing derailer is low-clamp/top-swing or high-clamp/bottom-swing, presuming it seems to be the correct choice for the bike (no frame interference issues and correct amount of gap between the cage and the teeth of the large ring). Some bikes can accommodate either.
Finally it's not a bad idea to double check that the new derailer will play nice with the ring profile (tooth count) of the large ring, the max total ring size difference number, and the minimum difference between middle and large rings. Of these, the first two sometimes give you some leeway to fudge it, and the last really does not. Mismatch the optimal ring profile a little and it will usually still work, and exceed the max total difference and the only consequence is it will rub the cage in small-small combinations. If you go under the minimum difference between middle and top spec, what will happen is the inner cage will hit the middle ring as you try and shift onto the large. You can try to get around it by raising the front derailer, but then you wind up with too much gap and you can have chain drop issues. So that's a good one to be hardline about. You can get these numbers with the above techdocs.shimano.com/google procedure. For example here's the sheet that has them for your existing derailer. This is all less relevant here because almost all 9-speed mountain fronts will be fine for your bike on these points, but other times it matters a lot.
The model code doesn't give you anything directly about the compatibility. Stare at Shimano model codes long enough and eventually you'll just know them or you can get an idea from them about chronology and quality, and that's it.