if a client indicates on my personal profile, "Book Again", does that count as an actual repeat client?

I'm not sure I understand your question. Where in your profile are you seeing this? If you are in your Inbox, looking at Past Stays, you would see a Book Again button on the right side (underneath the green Review This Stay button). The Book Again button allows you to initiate a stay request on the website (from a previous client who wants to hire you for another stay).