I think there are far too many conflicting rules or interpretations of rules for betting. First of all, I'd like to say that if action were head to head, it would make your situation more acceptable (to me). I don't like players to be allowed to push their stack (with or without a betting line) in a forward motion, and then allow them to withdraw, or change their mind and raise at the last second. How do you "release" three stacks of chips that are pushed forward and never lifted off the table top?
I think the rule is designed for players that lift their chips and move forward (as opposed to sliding on the table). Some rooms allow players to advance forward with a stack but do not count the bet until the chips are "cut and released." They are allowed to return the remaining chips in their hand back to their other chips. I don't like those rules. I think that any player that has intentions of making a wager, should know how much he is going to bet, and once he moves his chips forward either by pushing, or with chips in hand, he should NOT be allowed to change his bet. There are some rules that will not allow more than one forward motion, even after you say that you are raising.
Your example is what I define as a string bet. Players need to make their bets and intentions clear for all to understand. Why would a player do that? Why not just push all of the chips forward, or announce how much you are betting? It's simple enough.