If your team members are being notified when a client books online, and not when an appointment is entered manually by one of your employees, chances are your Booking notifications are set to look something like this - see below
In order to change these so that team members are notified when ever a client OR team member books or changes an appointment in their diary, go to "Settings - Team Members" , select the team member's name and "Notifications" then change this to "by anyone"