I have the impression this is a workaround but not a solution. If you access your site via github.com, you want the content of readme.md displayed, so people know, what this project is about. If people access your website via github.io, you want the content of index.html displayed. I was under the impression, that the right way to do this, is to exclude README.md from your website by including in your _config.yml:
Now the problems seems to be, that this is not always working as it should.
FWIW, for those (other) newbs out there, I found that if you have an existing repo, what they don't mention in the GH Pages tutorial, is you just need to to go to settings in the repo, then scroll down to the Github Pages section, and then set the (master) branch (this is disabled by default), and save it.
This should create a page you can then goto in the browser.
EDIT: theme is optional - I left it off.
See photos for visual guide.