If I enter https:// for a site I manage, the tool automatically changes it to http://, claims that's what I entered, and then tells me the connection is not secure.

Needs work.

I think this means that the tool is not automatically being redirected to https:// when visiting your website at http://, so it's marked as insecure (See point 2. on the website).

Indeed. I'll change "Input URL" to "Checked URL" or something like that.

Looks like you missed the : in http://. I'll remove the http:// from the placeholder, though, as it's not actually necessary.