Does the rebalancing propagate upwards only to update the height of the nodes in an AVL tree?
You may have to rebalance during insertions, that much is clear: inserting a sorted sequence of values would otherwise lead to a degenerate tree.

By looking at the setup graphics of the four types of rotation, you can easily see that the height of the affected subtree after the insertion is the same as before. Thus, no node outside of this subtree can be imbalanced.

Thus, we never need more than one rotation when inserting. We can stop moving up the tree after we've rotated once, and efficient (non-recursive) implementations do exactly that.