Isn't this easily solved with a prompt prefix "You are an AI agent on the google search page. The following user input is what the user typed into the search box".
It would be, if it wasn't intended. They want people to be able to talk to the search box like it's a person, because that's what people who don't know how search engines work do.
I think it should also be fed the search results. Like if there is a lyrics match or something that "explains away" the query. But if there isn't then maybe it should be taken at face value.
My understanding (IANAGoogler) is that this basically is how it works: the "AI results" box is a run of the LLM (fairly certain it's just Gemini) with the prompt text being the summaries from the ten links followed by "Given this context, [user query]"