Thank you Loewchen, it turns out that I forgot Rule 1 of troubleshooting.

After a restart of my computer for good measure, it logged in successfully.

Thank you for your help, Loewchen and Koub.

Thank you for that nugget of information. I just tried logging in on factorio.com, and I logged in successfully. After clicking on the Mod Portal link in the top left, it led me to the mod portal, but the logged in status did not come with me. So when I tried to log in with the same info put in at ...

When the new website was put live, I tried to download a mod. It asked me to log in, and after inputting my details it took me back to the page for the mod I had wanted to download, but the "login" button at the top right stayed "log in" and the mod would not download.

I thought it might be an ...