So according to the rules, if you do a solo run in a public Server you must press every button or run is considered coop.

I just want to clear one thing up for when there is more than 3 people in a public server (which there often is) and the multi button comes into play. Obviously other people are also going to have to press the button so you can continue to progress the level, my question is whether or not a run is considered valid if you press every button but other people had to press the multi buttons also.

That's a good question. The rules for public servers have not been updated since 2017, so the buttons were definitely updated since then. I'll look into this.