No, it's not. But it's more or less a lost semantic battle going on for more than 20 years at this point.
I don't know what to tell you if you can't grasp that the lack of "role play elements" is precisely the differentiation between the two.
An action RPG is an RPG where the combat side of the game is controlled as direct action. A hack'n slash is a "RPG" (loosely) where the "action combat" is ALL there is about the game. No branching dialogues, no meaningful choices and consequences, no relevant interactivity with the scenario and/or exploration, etc.
Going by your definition we wouldn't even need the two different terms to describe/differentiate them, because both are games with action combat.
Unless you are one of these morons who think a couple of stats and a lousy skill tree are what defines "RPG elements" in any game.
Which would make basically any modern title in existence a RPG, incidentally.