When I say "email me," I don't mean "make something show up in gmail," I mean, send me a message over the email protocol. I don't use email as a to-do list. I use it for correspondence, often with people who use other clients.
My conception of email is client-agnostic, and I think almost everyone I deal with (admittedly outside of the tech/valley/startup inside) thinks of it the same way. I hear things like "I'll shoot you an email," or "I need to check my email," or "what program do you use for email?", none of which imply that the speaker intends to be talking about an app.
Texting isn't an app, the telephone isn't an app, email isn't an app, at least for many people.
Yes, but which client? I use mutt, my office manager at work uses Outlook, our accountant uses Gmail, and our advertising manager uses Yahoo.
The fact that we can all email each other seamlessly is inherently because of the common protocol, even though we don't think of it consciously when talking about email.