Hi Mike,
I'm not sure how changing it from a button to a link will solve your problem,
but either way, no - there's no way to change the Login button (which is really a Form Save button) into a link.
If it's any consolation, I've tested example 3 here and you can press Enter when on the password field, and the Login button does get triggered, so if that's not working there then perhaps more detail about that is required?
cheers
Bruce