when starting a new tmux session with the tmux command, the first window still has the default name.
The manual states:
If no commands are specified, the new-session command is assumed.
So bare tmux is equivalent to tmux new-session. You need it to be tmux new-session -n ''.
Create a wrapper script or a shell function that will pass the long command instead of sole tmux. Example function:
tmux () {
if [ "$#" -eq 0 ]
then command tmux new-session -n ''
else command tmux "$@"
fi
}
Note this simple approach will "fix" bare tmux but not e.g. tmux -S foo. Adding logic in order to properly identify all such cases may not be trivial.