[ci skip]
Addresses #2530. Not closing that though, since the behavior might still be tweaked.
The healer will still play their own animation, if applicable (behavior option 2 in
the bug report).
I also moved the animation to be played *before* healing, which is the same
order as in C++ code. In the previous order, the animation isn't played for
some reason.
Fixes#2295.
