I object to "warp core" in connection with TOS for two reasons. I think it's lazy - it's this way in TNG so it's always been this way! And it discounts any changes or advances in Trek tech. It's like calling the things on the wings of a B-17 "jets". Sure, it's propulsion and even a similar layout and position. But calling them jets is silly.
But really it's "keep your damn TNG out of my TOS".
Didn't NX-01 have a warp core, though?