I thought login 'with email' meant use my email to login

Here the user is surprised. He thought that email button meant to use the usual email address and password combination below.

Only after he presses it does he realize that it’s going to send him an email, where he can get a link to log in.

So you need to at least they have the word “send”, not “with”, before the word “email” on the button or something to make it super clear.

Sure you could argue that there’s a final button at the bottom of the screen. But that is usually covered up by the keyboard on cell phones anyway and the user has to do extra thinking even if it is visible.

I think discourse’s description of this aspect is very accurate. Perhaps an admin of your site edited this text:

“Skip the password; email me a login link”