A message is any piece of important information given to a specific person (or to specific people). Messages can be delivered in person, in speech, in writing, in an e-mail or any one of many different ways. Using a mobile phone's short message service is only one way.
Certainly your American colleague's usage is imprecise -- it could have been a message sent by carrier pigeon. But your colleague assumes that you understand "e-mail", presumably because that is how you normally send messages to each other.
- For Teachers