Thanks, with all of the mentions of tickets and inboxes and assigning chats, it turns out that a workflow is the only way to actually create a notification. There is no documentation on how to do this of course, but using the AI feature came up with a solution, use Property name is Record source and New value: Record source is any of Bot Chatflow or Conversations.
It won’t create new leads or update existing contacts, but at least I can get a notification and see the details in the Inbox and decide what to do next.