You're describing someone in a service industry being appropriate with a client they want to cultivate as a client. They have to sort of pretend to be friends.
You are expending waaaay too much energy thinking about this. Especially if you are not going to do your part in moving anything possible forward.
Leave it alone.
Get your hair cut, trimmed. Have a bit of fun while the stylist does some business level casual flirting. Don't assume it means anything.
Stylists and barristas and bartenders and fitness coaches are used to people hitting on them/falsely thinking they are interested.
If you are interested and want to ask him out - do that. Otherwise, forget about the whole thing. He likely has no interest in you other than keeping you as a regular customer (free trims are part of that - everyone offers it - it's nothing special - it is part of customer retention).