I haven't read through the whole thread and I, obviously, wasn't present at the meal. But, from reading the OP's first post, it just kind of stood out to me that the 12 year old, who had just gotten done with swim team practice (if I remember right from the post) meant for the appetizer to be part of her meal like a side dish ... not a plate to be shared ... and the OP was able to pick up on this.
I'm kind of of the mind that if the girl wanted an appetizer "for the heck of it", she most likely would have OK'd this with her mother while checking out the menu options. And yes, I would think that the mother would be within her rights to share that appetizer with the table even though her daughter was the one who wanted it. But again, it seems to me that the girl order the appetizer as a "side" to her meal and it should have been treated as such, even by the mother.
On a side note, it doesn't seem as though this was a very large group, so the mother was most likely aware that her daughter was ordering an appetizer along with her meal. It seems a bit strange to me that the mother didn't take a moment to question what the daughter was ordering. I can't imagine just letting my son order an appetizer with his meal without questioning him. If we are having an appetizer for the table, I wouldn't be letting my son decide on and order it. There would be a discussion as to the choice and who exactly was hungry for one. Again, the way the daughter ordered the appetizer without consulting her mother just kind of seems as though she meant it as a side to her meal instead of as an "official" appetizer, and as such it should not have been shared with the table. I have done this myself, although I usually take home part of my entree (due to the fact that I am not on a swim team! lol) A big bowl of clams (appetizer) with a chef salad is my usually meal at one restaurant, and no, I don't share the clams. They are not on the entree menu so the appetizer is the only way I can get them. If DH and DSons want clams they know to order another bowl. Oh, plus I get half a chef salad for lunch the next day. Yum.