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If anyone is confused about how the web server could possibly know the domain with which you connect, it has been a required field in the HTTP request header since HTTP 1.1. I remember this one baffling me when I was younger.
@DocRoot it proves identity in the sense that it proves that the content is coming from the real owner of the domain. It does NOT prove who actually owns the domain; for that you need extended validation (and even that has a few difficulties).
"It would normally only prompt with this if you are allowing users to login (submitting username/password) or submitting personal information over an unencrypted connection. General "text" would not necessarily be a problem." This will soon no longer be true; Chrome will show the warning for ANY text entry field. It can't tell the difference between potentially sensitive information and benign stuff like searches, after all, so the decision was made to just warn on everything.