How can I identify the cause of this SSL error in Safari 6?
Safari 6.1.6 supports only TLS 1.0 (which is increasingly being turned off by servers concerned about the BEAST attack), and there's no error message apart from that generic "SSL error has occurred" in the Console.  You'd need a newer version of macOS to carry on using Safari on those sites.  If upgrading the OS is not an option (for example if you still need the older version of Xcode) then you could try Firefox 45.9.0 ESR, which uses TLS 1.2 (confirmed with www.HowsMySSL.com); that version of Firefox passed its end-of-life in mid-2017 (and more recent versions require a newer OS), but at least it's an improvement over Safari 6.1.6.  However I'd be more comfortable installing a modern OS alongside the Lion / Mountain Lion, or at least using VirtualBox to run a modern version of GNU/Linux inside it, and using an up-to-date browser from there.