[pmjordan]

You can transparently proxy arbitrary network traffic. See https://github.com/apenwarr/sshuttle for an example. FWIW, I'm only pointing out that HTTPS wins you exactly nothing compared to HTTP for this particular purpose. The concept of the "correct" IP address in the context of NAT and proxies is hazy at best anyway.

[billpg]

There are ISPs that will route all (or some) port 80 traffic into a bank of caching proxies but route traffic for all other ports normally.

(GRC.COM's 'Shields Up' service makes a point of using HTTPS to make sure it's testing your external IP rather than a random proxy server.)

I suppose it depends on what you actually want to use the returned IP for. As you say, the concept is hazy.