Just had a conversation about this with a Japanese guy. He says メール means both SMS and e-mail, says to send SMS in Japan you need the same contractor between the 2 phones, when you make a contract they give you a メールアドレス (e-mail address) in the form of xxx@xx.xx, SMS and calls are done using the same phone number.
Can we put a rest to this and call the boss’s number a phone number?
Edit: Addendum: “but after the advent of smartphone you can always use SMS regardless of the contractor”.