Water leaking when shower is turned on
Since the shower head is above the top of the tub, while the tub spout is below the top of the tub, it makes sense that a leak between the surround wall and bathtub only happens when the shower is running and not when the tub is filling.

It sounds like the caulk in that area is either failing or has failed completely. You'll probably want to remove the tub surround and pull out the damaged drywall again. It's also possible that there is a crack in the tub surround itself, so while you've got that pulled out, you may want to inspect that for damage and replace it as necessary - I'm not sure that you could effect a good repair on it, though someone may offer some tips for doing so.

When you reinstall, use cement backer board (one description) instead of standard drywall as this is designed for wet areas. It may not be fun, but I'd suggest replacing all the drywall behind the surround with cement board instead of just the one little spot. In the long run, this will be a better option and future you will thank current you for saving future time.

Once the cement board is up, reinstall the surround then caulk with a good quality, waterproof caulk. Make sure you do a good job of it, not leaving any gaps and doing everything you can to avoid leaving voids where water could puddle.