Read about the differences here:
https://mainstitute.org/blog/betrothal-versus-engagement
Okay, that's using a particular religious definition of the word, so it's not an objective source. The dictionary and Wikipedia say that betrothal is synonymous with engagement, or at best a subset, a type of engagement negotiated between families with or without the consent of the betrothed parties. So not all engagements are betrothals, strictly speaking, but all betrothals are engagements. So the distinction doesn't work here; if Spock was betrothed, then he was engaged.
The whole premise of the Galileo Seven episode being Spock's first command has been totally subverted by the S2 opener, where Spock is left in command of the ship, steals it out of spacedock, and averts a war with the Klingon Empire.
Just as "Memento Mori" contradicted Spock's line in "Dagger of the Mind" that he'd never mind-melded with a human. These kinds of adjustments are inevitable in a prequel, and they're no worse than the internal contradictions within TOS itself. Really, neither the "first meld" thing nor the "first command" thing makes sense in light of "The Menagerie" establishing that Spock had served with a human crew on the Enterprise for 13 years before TOS. They're just early-installment weirdness like "Vulcanian" and UESPA.
The opening teaser of TOS S1 The Corbomite Maneuver also shows Spock in Command of the 1701 while Kirk is in Sickbay for his quarterly physical check (and McCoy is making damn sure he's not interrupted too.) Spock gives orders to BOTH Bailey and Sulu and at the end of the teaser has even ordered a Red Alert.
That's not a good example, since having the conn while the captain is off-duty or indisposed is a different thing from being the sole commanding officer of a ship or mission. It's not really "a command," it's just being a proxy for the actual commander.
One could also argue that taking command by necessity when your superior officers are captured is distinct from being formally assigned command, but that may be too nitpicky even for me.