However, the very term "impulse engine" implies propulsion via Newton's third law. Hence, the original conception of them as nothing but rockets, possibly using nuclear fusion as the reaction.
That's the general impression I've gotten for years now. The "atomic matter piles" referred to in the TOS episode "Court Martial" as existing aboard the Starship Republic
might very well have been connected to the ship's impulse drive, plus there've been a few times in the history of the franchise when sublight impulse propulsion was referred to as using nuclear fusion reactions. To be frank, I've accepted Starfleet impulse drives as being nuclear in nature since I was very young and learning what I could about the designs of starships in the franchise.