Wow, holy smokes, that's a manufacturing defect if there ever was one. As XevetS said, the bottom pic has the cable installed correctly.
As for the "connector" itself, as you discovered, it's not really a connector. I suspect the purpose was simply to speed manufacturer by being able to pop in and solder all the wires at once via the housing vs. soldering in each wire separately.
The insulation stuff on the bottom of the board isn't really bonded, it's just a fairly strong adhesive that comes off mostly clean (at least it did on mine). I removed that connector thing on one of my Solos and soldered one of my cables directly. What made it challenging was A- removing the pins left when the connector broke off, and B- the solder they used in manufacturing didn't melt very well. It was probably lead-free, which tends not to melt as easily as older leaded solder. Even with a good iron and lots of heat, it just didn't seem to want to melt or stay melted.
It's doable if you have to and if you're decent as soldering though, but yea, I definitely advised Rich that it wasn't worth the effort. But in your case, if it's your only option, that's different.
If the connector was in tact, I could have made you an extension that reversed all the wires. If you think you can de-solder it, PM me and I'll hook you up with a cable, I'll just make it a few inches longer than my little extensions and you can solder it right to the board.
If you trusted me enough (acknowledged that you have no reason to!), to sent me your main board, I'd happily solder on the cable and send it back. Gotta help each other out, right?
If you do remove the board from the Solo, be real careful with that gray accessory port ribbon cable. Several wires broke on mine and it was a royal PITA to fix, just as a result of having to completely remove it, strip back each wire and solder it back on... again with the cruddily melting solder they used. Just don't bend it back and forth much or at all. 2 or 3 times was all mine took for the wires to start breaking.
