Did you coin the term "pro side"? Lol. That's my point, that NO ONE knows what it means except maybe the person who coined it.
Everyone has their own opinion or meaning. But I do think your definition of it, playing the ball to the side that will leave it safe if missed, is the most common opinion.
I think we just differ on the intention of aiming a shot like that. I believe the intention is to give you the best chance of making the ball. If the shot is accidentally overcut or undercut, and it misses the pocket and leaves the opponent tough, people say..."He missed it on the pro side".
And that only makes sense if the "pro side" is (as Max Eberly explains) the side or portion of the pocket that gives the player the best chance of pocketing the ball. Where the ball goes after missing the pocket is an added bonus for being able to at least target a specific portion of pocket opening.
In other words, a ball can miss to the left or right of the intended aim/target. If the player is aiming for the most accessible portion of the pocket and misses, it's better to miss outside of the intended aim rather than inside. When we miss outside of the intended portion of the pocket we're aiming for, the ball doesn't generally hang up and stay in the pocket.
Nevertheless, the intention was to hit the pro side of the pocket in order to pocket the ball, not in order to leave the ball tough in case we miss it. That part of the shot just comes with the territory, with the skill of being able to aim the ball into a certain portion or side of the pocket that gives the best magin for error. Most amateur players don't have the skills to aim that accurately, which explains why it's called the pro side of the pocket.