If you boot up the Live CD again, does the battery start charging? Have you tried resetting CMOS? (Disconnecting and reconnecting the CMOS battery after 5 minutes or so)
Ok, I reset the CMOS (disconnected, reconnected CMOS battery - and on next reboot it told me the CMOS checksum failed, so indeed it got cleared, and that it took on default values). I just set the boot order correctly and tried again.
1) Still, the "fried" battery neither charges in Windows nor in Ubuntu. Furthermore, in Windows the Thinkpad Battery manager or whatever shows "Charging" with 0 Wh energy out of 47.6Wh full capacity, 11.23V battery voltage and 0.00A current. I.e. it's trying to charge but zero current is flowing. It's still amazing to me that the (Ubuntu) system could, by discharging the battery, fry it. BTW, the battery manager in Win tells me that it's actually on cycle 76. Anyway, it's pretty new.. and worked perfectly, charging up to pretty much the full 47Wh consistently, before the fateful time I tried the Ubuntu LiveCD without wall power.
2) Going off this battery topic to another issue.. about my CMOS resetting. Weird, after I reset the CMOS, I would actually get stuck at the F1/F12 screen (go to BIOS or choose boot options). I could go to F1 (BIOS), but F12 would freeze the machine, and not pressing anything would also not start the hard disk boot (even though it is first on the boot list, it worked fine before, etc). I was really getting worried that I would never be able to start the machine again from my HD, since I could only get into BIOS, but not boot (from HD or CD). But then I forced the system to do a full memory test and full diagnostics on boot rather than quick boot, and it then booted to HD (Windows) or CD (Ubuntu LiveCD). Then, setting the settings back to normal, it keeps working as it did originally, so everything's fine again. Somehow it got its settings happy again.
I have in general had the experience that my A31 is very flaky, though I love it, whereas others here have reported them to be very robust. From the kind of problems (sometimes not waking up from sleep though rarely, taking forever to realize you're trying to wake it - even needing to press keyboard keys to wake it from suspend on occasion, etc)... I suspect the BIOS. I have the latest BIOS/embedded controller firmware, but I wonder if I should stick with the old versions, i.e. downgrade back to the version that was originally on it. I'm concerned that the new BIOS may be for other things, and that the backward compatibility to my A31 may not be the priority right now since they're so old..
Anyway, about the battery issue.. I still have no clue what's going on. Is there a battery charge cct on board? Is it possible to reset it? Is there a pinout online for what the 4-6 contacts of the battery do, so I can see what's wrong with it with a multimeter, etc.?