How do I change the log in button TEXT?

We use the language file system of Joomla, and what you describe is unfortunately a limitation when trying to create multiple instances of a module.

All I can suggest is to use a module title for each module with the text you describe. The button would just say "Login" then.

I hope that helps give an idea of some possibilities, but if you need something else, just let us know.

Thanks,
Alex