Creates a booking for the given appointment type and slot. Only free appointment types can be booked through this API; a paid appointment type (see pricing_type on the appointment type) returns 400, since paid appointments can only be booked through the site's cart and checkout UI, which collects payment first. On conflict the API returns 400 (the host already has a booking at this time or is not available); the agent should re-query /availability and pick a different slot.