Nope, them being "rockets" by any stretch of the word is the thing that is completely ruled out.
Except that that's exactly how Rodenberry envisioned them in the TOS writer's guide, and how Okuda and Sternbach envisioned them in the TNG tech manual. We even have a very good description of how impulse engines basically work in "Emissary" when Chief O'Brien uses DS9's shield generators to create a subspace field, effectively converting the stations attitude control thrusters into low-powered impulse engines (allowing it to travel a distance of several AUs in under 24 hours). This for a space station that is not even supposed to be capable of long-distance flight; this gives us a very good indication of what "thrust" can do if you deliberately screw with the laws of physics, as Starfleet does on a daily basis.
As said, impulse engines are used -- and depicted -- inconsistency, but then so is EVERYTHING on Star Trek, and in this particular case there's not a lot of wiggle room for how they work. It's a combination of "thrust" and "other stuff." It's the "other stuff" that varies from ship to ship, engine to engine.
If you really still want to argue "It's completely unlike a rocket and meets none of the criteria but that doesn't mean it isn't a rocket" because you like the word rocket a lot
I'm not arguing anything. I'm flat out SAYING that impulse engines are a rocket-like propulsion device because the producers of the show said they are. There are a lot of reasons why this doesn't make sense from a real-world point of view, but then if Star Trek were realistic it would totally unrecognizable to begin with.
Whether Spock was speaking of impulse engines when saying "rockets", that we can argue.
No, we really can't. This is a line from "The Cage" at which point Rodenberry hadn't worked out the more advanced terminology that went into later episodes of the series. Much like your own analogy: "Phasers" were called "Lasers" and "deflectors" were called "meteorite beams" and "warp drive" was called "time warp" and moved the ship by "time warp factors."
So we can either conclude that Pike's ship was carrying a set of bizzarely anachronistic technologies for some reason -- including conventional rockets in the impulse deck and handheld laser pistols -- OR we can just chalk it up to the language barrier. Say, for example, that most of Pike's crew actually speaks German and the universal translators are way overdue for a firmware upgrade (I even played with this in a fanfic recently; when the Keeper actually speaks instead of transmitting his thoughts, the translator gives the most literal rendition of his words and he repeatedly refers to Enterprise's organization as "The Federal Sunfleet.")
Out of universe, though, we consider that we know what the writers were actually referring to in those cases, and precisely what role in the story those plot devices actually serve. That can be reconciled without introducing any new speculative technologies from our end.
Is that "reaction control", though? The rockets never flared in order to turn the ship around - they were exclusively used for lift, in both movies.
No, their counterparts on the upper saucer fired briefly in orbit of Vulcan when the Enterprise was dodging through the debris field. This follows Pike's "drop us down underneath him, Sulu!" order.
Worth considering that the term "RCS thrusters" is not a canon reference to anything on a TOS or TMP era starship. They were only ever referred to as "maneuvering thrusters" and ST09 and STID are both very consistent with that.