Please explain the goalkeeper back-pass rule in detail to me...

I have noticed in the amateur soccer league that I play in that apparently goalkeepers aren’t allowed to handle passes kicked directly to them by teammates (back-pass rule, just like FIFA) but seemingly if they first stop the ball with their feet, and kick it gently around a bit, then they are allowed to pick it up with the hands - is this a violation of the rules that has gone unnoticed repeatedly, or is this actually what the back-pass rule permits?

Edit: Assuming FIFA rules apply (every league may have slightly differently rules, but just asking what would happen if a keeper did this in a FIFA rules game.)

It’s not allowed. An indirect free kick should be awarded to the other team, just as if the keeper had picked up the ball immediately. Otherwise it would be a loophole that would defeat the point of the rule (which was to prevent it being easy for teams to waste time by simply passing it between a defender and the goalkeeper).

Specifically: Law 12 (Fouls and Misconduct) states:

It doesn’t matter how long it takes him to pick it up.

ETA: If you are in America, Velocity, sadly, there are a lot of rather stupid applications of the rules that one sees here, fueled in large measure by the fact that most referees never actually played the game at anything more than a very, very amateur youth level, and don’t watch much of it on TV, either.

OK, thanks. Now is it correct that the back-pass rule doesn’t apply if the ball last made contact with a teammate but it wasn’t a deliberate pass to the keeper? (i.e., opposing striker takes shot, ball caroms off of defender’s legs, then goalkeeper picks it up?) Or, if the defender somehow causes the ball to move towards the goalkeeper (i.e., trips over it) but not with intention to pass back?

Correct, when there is no deliberate pass or it is headed or chested to the keeper then it is fine.