TMP makes the same implication with an impulse engine.
That's why to it makes more sense that the first warp engine was actually an impulse engine since warp and impulse essentially are both types of field propulsion.
It tallies nicely with Balance of Terror
too - for the episode to make any sense, the Romulan vessel had to be capable of FTL travel and it has what appear to be warp nacelles, with Scotty describes their power source as "simple impulse". Impulse Power could very well power the warp field machinery.
For this reason, I think "impulse" is better described as a source of power rather than a form of propulsion - it's just that propulsion is what impulse is commonly used for.