General flash video recording
URLs starting with rtmp:// like in the following example:
rtmp://somehost.com/somedir/somefile.flv
use RTMP protocol to stream flash video .flv files. As you can see you don't have to be a rocket scientist to know that

When recording any stream you usually start with a URL finder software even if it is built-in into a stream recorder like Replay A/V or Replay Media Catcher.
'coz you already have a couple of stream recorders, I suggest you to use free URL finder software called URL Snooper.

When getting flash URLs try to find ".flv", ".swf", "rtmp://" in the URL.

Recording flash video from eonline
The web-site you have mentioned seems not to be using HTTP or HTTPS.
Although after disabling TCP connections via port 1935, TheVine player connects via HTTP port 80. You can see server POST and GET requests with Proxomitron, but I doubt that you can get any useful information from it.

The following string was mentioned in the POST data
so TheVine supposedly uses the RTMPT protocol (RTMP through TCP). The URL Snooper can find such URLs if you add "rtmp;.flv" to the "Also search for" textbox.

Although theVine @eOnline developer claimed to use Akamai (FCS1.7 server) I can't watch it. The video keeps loading and loading for an eon.

You can save the stream with unique stream ripper called Replay Media Catcher. If you don't mind to lose some quality you may record any streaming video with screen video capture software, but of course stream ripper software gives much better results.