Some responders talk about soldering the "socket" (the tiny white plastic piece with the movable brown clip) to the logic board. On my 80gb Classic the plastic "socket" actually gets inserted to a female socket which is itself on the logic board. When I replaced my original battery recently I failed to get the battery strip lead inserted properly. I opened and tried again a couple more times unsuccessfully. On the second attempt the "socket" came out of the logic board. I eventually got a close enough look at the arrangement. I then removed the socket from the logic board and inserted the battery lead into it. I then used a toothpick to seat the socket in its receptacle. I still don't think I have the brown retainer seated properly but everything works.
I can only assume that Apple may have had the socket soldered to the logic board and somewhere along the way got away from the soldering and just used the good old male and female plug/socket method. IF my battery lead eventually works loose I will probably open my iPod back up and either push the brown clip down properly or use a very small amount of super glue to hold it to the socket. To confuse owner repairers Apple uses two different socket arrangements for the two leads iFixit covers in the repair. The Battery clip is an up down arrangement while the "other' socket uses a 90deg flip. For battery replacement purposes I don't see any need to remove any lead other than the one for the battery.