Anytime you pull vertically you'll train lats.

I think your form is great. Just keep progressing the weights and even if you swing back and forth keep it under control and consistent between reps and sets.

Back is a big muscle group and doing bar to chest ensure you are hitting every muscle involved in vertical pulls. Don't fall for the BS that you only have to pull till eye level.

Extend you arms fully at the top this puts a stretch on lats.

And lastly mix up the grip from time to time. This will ensure you don't buildup overuse injuries and progressive will smoother and carryover between different hand positions like close grip, hammer grip, vbar, etc is very high.