A short thank-you is appropriate if the recruiter gave you useful time or information, but it does not need to be formal.

The point is simply to close the interaction professionally. A two-sentence message is enough.

You can thank them for clarifying the recruiting timeline and mention one useful point from the conversation.

Send a concise note the same day or next morning, then move on unless there is a clear reason to continue the conversation. If useful, this guide on consulting coffee chats gives more detail.