On .tmux.conf, what is the difference between "bind" and "bind-key"?
"bind" is an alias for "bind-key", they have the same effect.
Also you can shorten commands as long as there is only one command that starts with the letters you write, so for instance "bind-k" or "bin" would work as well.
From the man page:
bind-key [-cnr] [-t key-table] key command [arguments] (alias: bind) (...) Most commands have a shorter form, known as an alias. For new-session, this is new: $ tmux new vi Alternatively, the shortest unambiguous form of a command is accepted.