Adding
export EDITOR=vim
to your .bashrc file should really do the trick. (Quotes aren’t necessary there and, depending on what quotes you used, they may be the cause for your problem.)
You must open a new shell (or enter source ~/.bashrc at the prompt) after modifying file .bashrc for the modification to take effect.
What is the program from which you want Vim to be started?
I haven’t used Git, but the documentation reads:
The editor used to edit the commit log message will be chosen from the GIT_EDITOR environment variable, the core.editor configuration variable, the VISUAL environment variable, or the EDITOR environment variable (in that order).
So check whether one of these variables is set:
echo $GIT_EDITOR $VISUAL $EDITOR
git config --get-all core.editor
For me,
export VISUAL=vim
solved the problem.