Typically you would create an account that uses 'User Name' (and if you want, you can do a force caps/lower case in the page so it doesn't matter how a user types it). That basically implies that if you are using an email as user name, the user should just type in their email.
You are asking for problems if you let them login without the email provider, because you won't be able to distinguish user:
abc@gmail.com vs. abc@outlook.com etc.
