After reading about method 2, I just had to give it a try. I recently dug up my old Pokemon games. My Blues, Red, and Silver are all fine, but my Crystal died. It hasn't quite been 12 hours yet, but so far it seems to be working.
I have an old Gameboy Color and, way back when, had bought a couple Shock N'Rock packs for it, which came with an AC adapter that also fits right on the GBC. I plugged my Crystal into my GBC, hooked up the AC adapter, made a new game, and saved just after stepping out of the house. I must have bumped the cord at some point because when I brought it back out to test it, it was going through the opening sequence (the cord/connector on that old AC adapter is a bit touchy) but my save file was still there. I turned the GBC off, unplugged the AC adapter, and waited a couple seconds. Then I plugged it back in and turned it back on, and my save file was still there, which was an improvement compared to tests I did when I initially found out my old save game was gone. I put it out of the way and am letting it continue charging. (And since you don't have to save when it's done or anything, after 4 days or so I'll just shut it off, wait a while, and turn it back on again. And hopefully start playing Crystal once again. Otherwise I may be looking into buying a soldering iron. *image removed*
FYI, Lithium-Ion and Lithium-Polymer aren't the only rechargable Lithiums around, just the most recent. Not really sure if this battery is meant to be rechargable, but if you think about it a rechargable battery would make a lot of sense. As long as the user plays the game a lot, the battery should keep getting recharged. Of course, even rechargable batteries will fail eventually.