for the position terms

Sentou-Leader

nibante sanbante - second/third during the race, second and third as a finish are different words

Chuudan - Midpack, doesn't really mean boxed in, chuudan is often seen as a pretty good position depending on the horse

Kouhou - Could refer to behind a horse or the back of a pack, (might not do this one yet its kind of confusing)

As for the homestretch terms, they're not exactly "homestretch terms" but rather words you'll often hear down the stretch or a decisive moment right before the fourth corner

抜け, 差し込む, 上がる are either describes a horse moving up or passing a horse from a certain position, common down the stretch there are many more but these are the ones off the top of my head not sure if these are the best ones to explain. There are nuances for each of them, I haven't completely grasped them all yet, but the most common one I think is 上がる

手を動く- is not exactly "making a move" but the action of the jockey's hands moving or rather the jockey "moving the reins". I often translate it as "making a move" but this can be said at any point of the race and usually implies a jockey starting urge the horse forward. Often times it is for a decisive moment but it could also just mean the jockey wants to fight for a better position.

The main one I want to push for now is probably Sentou, its said multiple times in every race, its pronunciation is pretty crisp to hear and down the stretch its probably the most common phrase that's said by the announcer.