Issues with horizontal pressing are often different than vertical. In other words, what causes issues in a bench press is often different than an overhead press.
In benching, people often get sensitive to the bottom of the motion, as the elbow moves past the shoulder. Pain is common in the front of the joint. The isn’t typically from impingement, at least not in the colloquial sense of the acromion pinching the supraspinatus.
Think of it like a lever. As the elbow (bottom of the lever) moves back, the head of the shoulder (top of the lever) will at some point start moving forward:
(This can cause impingement of the biceps tendon, but that’s not typically what people are referring to when they use the word.)
An easy way to get around this to still be able to press / work the chest some is simply limit how low the elbow can go. Floor Pressing is my favorite substitute:
You just don’t let the elbow go that far behind the shoulder. If that doesn’t work, then you can place a pad under the elbow to really limit how much range of motion the shoulder gets.
As far as overhead motion, it’s often not so much what you do but how you do it. For instance, you seem to have a good deal of sensitivity to internal rotation with overhead motion- as you raise the arm, if you let your palm turn in (usually necessitates internal rotation of the shoulder), then you have pain. If you don’t, then you feel better.
You’ve already hit on some substitutes for this then- use a neutral grip.
The other aspect to appreciate here is what are you doing during the day? For example, if you mouse in a way that’s constantly bringing your shoulder into extension:
then that could be one reason the shoulder is so sensitive to extension with exercise: you’re already doing the motion a lot.