I know from experience, personal and otherwise, that a push/pull setup can be very confusing at first because a lot of exercises get muddled and it's hard to tell whether they are a push or a pull. It really is quite silly and I think you nailed the take home message "It matters not whether the lift is a push or a pull. Your body doesn't care. Instead be aware of the actual movements taking place and the actual goal of the lift."
I'm not a huge fan of push/pull setups to be honest. I'd much rather use the theme of upper/lower, even then it is a very loose organization, you're still selecting exercises and movements that accomplish a specific goal.
BUT, no matter whether you use push/pull or upper/lower, in very rigid terms the discussion can end up becoming very ridiculous. Is a deadlift a push or a pull? Is an overhead squat an upper or a lower? Does the deadlift go on back day or leg day? Only bro-science can help us now.
The strength trainee says "Why sacrifice intensity when I can sacrifice volume"
The bodybuilder says "Why sacrifice intensity when I can sacrifice form"
"We are not sport, when there is a sport issue, we are not so good. The boxer is much better than us at boxing. But he will have to protect his balls if he wants to punch us."