As Andy commented, some Android stock browsers are based on chromium.
The issue you explained is kind of a bug where Android WebView built on chromium fails to loads the page in question due unmapped ssl error.
As explained here: https://bugs.chromium.org/p/chromium/issues/detail?id=395638;
Android Kit kat WebView component built on chromium is unable to open
web page due to an unmapped ssl error! It is usually occurring on
secure login pages presented by captive portals.
Here is their extracted log entries
04-09 12:49:49.215: W/chromium(24516): [WARNING:ssl_client_socket_openssl.cc(182)] Unmapped error reason: 261
04-09 12:49:49.215: E/chromium(24516): [ERROR:ssl_client_socket_openssl.cc(870)] handshake failed; returned -1, SSL error code 1, net_error -2
04-09 12:49:49.245: E/WiFiWebLogin(24516): ExtWebViewClient - ExtWebBrowser-veiw-1 - onReceivedError: errorCode=-1, failingUrl=https://126.96.36.199/login.html?redirect=syslynxservices.com/ description=net::ERR_FAILED
04-09 12:49:49.265: I/chromium(24516): [INFO:CONSOLE(12)] "Not allowed to load local resource: file:///android_asset/webkit/android-weberror.png", source: data:text/html,chromewebdata (12)
Had the same issue, and what stuck me is that they sugested it usually occurs on secure login pages by captive portals which is right where I saw that funny message telling me "we have detected a you are using an older version of chrome".
Could'nt come up with a solution but had to switch browsers but like you noted the issue vanished away eventually.