I admit that this particular error seems to have been reported on-and-off over the years. I too checked back through Google and found the threads that you just identified. Some of these seem to be associated with the introduction of WP 3.8.
However, I’m at 3.8.2 on all my sites and have been for a while. The only plugin I was updating was WF to 5.0.3. I have since updated another plugin (Google XML site maps) without any similar result. Some of the other users that noted issues before discovered that they had failed to complete the configuration of their WP site in the General settings area; e.g., no entry for a site URL. My sites have been running for years and have no missing entries. Also, no proxy is involved. I’ve never seen this particular error on any of my sites before. I’ve got about 8 active right now.
I might add that I went and tried disabling the XMLRPC option on one site before I did the update. This seemed to have no impact on the arrival of the errors listed above; that is, they still occurred. I’m going to see about deactivating WF before I do the upgrade and see if that makes any difference. I believe I’ve done three or four sites now, with four or five to go.
The good news is that I cannot discover any negative result as a consequence of the warnings. I also haven’t checked update.php to see what happens at lines 120, 272, and 417. I’ll also observe again that the warning refers to an inability to setup a secure connection to www.remarpro.com, and that this is post an openssl update to avoid the Heartbleed flaw. The sites that I’ve updated do not have an SSL certificate; however, one of my sites did, it had the flawed version of openssl, and I updated it per the instructions on the Internet through WHM and “yum update” and a site reboot. Having said that, I assume that the secure connection referred to in the warning would have been an ssl socket established at the www.remarpro.com side. What might be a hypothetical connection here is some interaction between a reset to XMLRPC and re-establishing openssl to www.remarpro.com, arguably after their certs, credentials, or other link parameters have been modified.
So this may amount to nothing. It’s just that I don’t imagine that I’m the only one seeing this warning, and it might be worth some effort trying to figure out what’s behind it.