Is there some rule that says they must declare a filibuster before voting begins?
Good question. As I understand it, they call for a vote for the end to debate, and if that vote fails, then the filibuster is in effect. I think at that point, someone, perhaps the parliamentarian, declares that the issue of the Gorsuch nomination is in filibuster status.
If the vote to end debate gets at least 60 votes, then they proceed to a final vote on the Gorsuch nomination.
A filibuster is a refusal to end debate and call the question. If the vote never occurs, you never lose. The cloture vote exists, on the issue of calling the question - and a cloture vote can be called any time (after, IIRC, a preset minimum time). But the cloture vote fails if you dont get 61 votes.