Is "Add AI summary" the same as "Summarize thread"?

No. "Summarize thread" on a ticket adds an internal recap to the thread for your team; it is never sent to the customer and never quoted in replies. "Add AI summary" in the reply editor writes a recap addressed to the customer and inserts it into your reply, so it is meant to be sent. Both are written by the AI in the ticket's language; only the reply recap leaves the workspace, and only when you send the reply.