The exact description of the congratulations offered was,
Let’s assume that even without exact quotations for all parts of the exchange, this is an accurate rendering. (That may not be the case, but that’s the text we have to work with.)
Notice that Zara Phillips herself apparently refers to a medal win. She acknowledges, in the next breath, that the word “win” does not always refer strictly to first place. Given that this was an Olympic medal, I think Tim Walker was colloquially correct to refer to it as an Olympic [medal] win.
So I stand by the reckoning that the form of her “correction” was inappropriate given that he made no explicit error in the description of the event for which he was offering congratulations.
Anyway, “get your facts right” comes across as ungracious at best even if he had made an explicit error. She did have a medal to be proud of, so the proper form of response would have been, “thank you, but I’m afraid it is only a silver medal; the overall winner was --.”