Possibly a common nickname that had been in circulation for some time that they decided to make official? Pike could have been making a joke that it had already caught on but that Starfleet giving it official blessing would make it "uncool".
I like that.
We see Archer in Enterprise pondering Starfleet adopting some sort of "Prime Directive" so chronologically the phrase has already been used way before the 2250s, though not in the same specific context.
He said, "some sort of a doctrine", then referred to it later as a "directive", no "prime".
At this point, I have to take the show as a soft reboot, which is fine. It does seem strange though, that they're purporting it to be a prequel when there are mounting unnecessary niggling little call-forwards that sort-of-but-don't-quite tie in to TOS.
Spock's actions here undoubtedly would have come up in 'The Menagerie', when he does the same again, but everyone (including Uhura) is shocked and all, "There's no way Spock could do anything like this!"