The thicker conductor is the tinsel. It goes from the driver terminals to a rivet in the cone. The thin wire is one of the two lead wires from the voice coil (VC). The thin wire is usually fixed to the cone by a black adhesive (the thin strip you see on the back of the cone). The tinsel and VC lead are soldered together at the rivet in the cone.
Unfortunately, I cannot decipher your last two pictures. Are you saying that the VC lead broke where it goes into the voice coil? Is the mechanical and electrical connection good at the rivet?
If the VC lead broke at the VC former where it becomes the VC, then your repair becomes much more difficult. Since you were able to reconnect the broken lead momentarily, you should be able to solder them together. If a soldering iron tip can reach into that spot, you are golden. You will need to use flux to prepare the leads and then solder them together. It would be best if you can scrape off a bit of the enamel insulation on the wire first. A small knife blade should suffice.
If you cannot reach the broken spot with a soldering iron, the repair becomes more difficult. You will have to remove the outer edge of the spider from the driver frame and VC former so you can get access to the VC.
After the broken lead is soldered, you will need to replace the rivet in the cone, secure it in place, and then secure the VC lead to the cone. Hopefully, others know the best way to reinstall the rivet in the cone. I can think of a few ways, but I have never done it.
If the repair is beyond your capabilities or you not sure you can do it, the best option may to acquire a replacement woofer and sell your broken one. I am sure there are some out there than would consider it a good challenge to repair your woofer.