You can see the result as a .gif if you click the image above. I am quite happy with how it turned out.
I wanted to mirror the design on Facebooks website, where you have all the sign-up information available right away without any clicks, as much as possible but needed one button to fit both the "sign up" and "sign in" buttons on the same screen initially. I also borrowed a feature from Slack, where you don't provide a password initially, but rather through a link that is sent to the e-mail you register. A neat little extra security feature in my opinion.