“Bring a pen or a book” illustrates choice between alternatives, a common use of aw. “Go up to the door” uses ila to mark an endpoint. Whether an endpoint is included in a particular legal application depends on the expression, context, and relevant rule.
Hatta can also mark an endpoint, yet it is not automatically equivalent to ila in every setting. A ruling about one particle or example cannot be transferred indiscriminately to every other case.