I wanted to close the loop on this for anyone losing sleep over the purpose of the button.
I didn’t see a pump. The owner of the shop I bring it to had told me a while back that he wouldn’t fail it on the state inspection for not having a windshield washer set-up, so I assumed he looked too. Therefore, I assumed there was nothing there.
I hesitantly ended up reaching out to the PO as I needed to know. Based on his response, it was for a washer pump and he forgot to mention it since he hadn’t used it in years.
I’m not going to try it until right before the next time I wash it, as I don’t want whatever is in there on the paint.
I truly wish it had any of the things in the replies, but I’m just not that lucky I guess.