Removing the battery was a nightmare. Clearly Apple did not intend the iPod to be repairable, and WENT OUT OF ITS WAY to make repair very hard. In my case by having the battery glued with two tons of glue. Eventually I found the following solution [but be very careful].
I inserted a small flat screwdriver from the top, going under the piece logic board [to the right in Figure 6], and very carefully inserted it under the battery and twisted it so that the flat blade lifted the battery. Do not stick the screwdriver far in the middle because there there are connectors to the wheel, but on the sides it is OK [and there is where the glue is more likely found].
This is absurd. One small drop of glue would have done the job of keeping the battery in place. But Apple added glue everywhere they could. They would have welded the battery to the case if they could.
Idea for Apple: place scorpions inside your devices, so when the user tries to open them they bite. You know, like in the mommy movies.
Removing this connector was darn hard, specially since I did not want to damage the logic board. Eventually I used a bit of WD-40 to loosen it up, and it came out easy. Hope I did not ruin something with it.
Just to make this idiot proof [e.g.: folks like me]. The right edge of the iPod means this: place the iPod face up, with the screen at the top and the wheel at the bottom. Then right is your right [note that it is your right, not the iPod's].