I think Github have already won the cultural dev war. I hear many a developer saying Github instead of "Git".
A recent example from a mid level dev:
"I'm gonna branch from the github repository" - when the repository is self hosted
A recent example from a mid level dev:
"I'm gonna branch from the github repository" - when the repository is self hosted
Docker too. ('a docker')
If that keeps up you can just call your alternative repo storage system GitHub 2.0 or GitHub Enterprise or something.
Github != Git