^ Well, I'd say that wasn't really clear either. That exchange in "Journey to Babel" went as follows.
Kirk: Mrs. Sarek, I just don't understand.
Amanda: Amanda. I'm afraid you couldn't pronounce the Vulcan name.
Kirk: Can you?
Amanda: After a fashion, and after many years of practice.
This might suggest a family name to some, but I never thought so. For one thing, Amanda tells Kirk that he "couldn't pronounce the Vulcan name" after asking him to call her by her given name. I thus assumed that the "Vulcan name" of which she spoke was an analogue of her Terran given name. For another thing, nowhere in the original series--nor, I believe, in any of its scions--do we learn of any Vulcan definitively having more than a single name. That doesn't mean that they don't, of course--it certainly took long enough for Sulu and Uhura to be granted given names--but I always found it compelling enough.
The thing is, for episodes like "This Side of Paradise" and "Journey to Babel," it has always seemed clear to me that the writers were playing with the notion of one-named Vulcans. They played with Spock having "another name," but neither offered the name nor even established with certainty that he even had "another name." One can imagine any number of in-universe explanations for Leila Kalomi's question of Spock. Back when they knew each other on Earth, where she had fallen in love with him, perhaps their conversations had turned to Spock's life, and he had intimated that he had "another name" without ever revealing it. Or perhaps he'd told Leila that some Vulcans took other names under specific circumstances or for specific reasons. There are uncounted ways such conversations could have set up her years later saying, "You never told me if you had another name, Mr. Spock."
The added (but minor) complexity of this might seem to fly in the face of Occam's Razor, but I feel the reverse is true: that Vulcans in general, and Spock in particular, having more than one name, despite hundreds of hours of film and television failing to produce one, seems an unconvincing complication and unlikely at best.