Sorry to keep going on about this, but I think I have isolated the location of the very mild pain in my shoulder. It is more toward the back of the shoulder. If I do a double biceps pose, I get a twinge at the back when I rotate the shoulder for the pose.
Thanks.
The only definitive test would be in person, but based on your symptoms, the problem is most certainly rotator cuff related, and at a guess, damage/inflammation to your supraspinatus. I would ditch overhead pressing entirely and any other exercise that irritates the area for the time being.
While it's still acute like this, you may even want to be cautious doing stuff like lateral raises (in particular) or external rotations. Hedge on the side of being overly cautious. I say this as a person who accidentally tore (a couple muscles of) his rotator cuff last fall, and had very few warning signs, and all of them were extremely minor. I.e. this stuff can sneak up and bite you far more quickly than you'd think.