OK. I'll stand by. Thanks for your help.
For reference, here's what apm tells me on the r197 system:
(in emergency mode (which boots with or without AC plugged in; this time booted without; battery known good and fully charged): )
# apm -V
apm version 3.2.1
# apm -vm
APM BIOS 1.2 (kernel driver 1.13)
Off-line, battery status high: 88%
(plug in AC:)
#apm -vm
APM BIOS 1.2 (kernel driver 1.13)
Off-line, battery charging: 88%
(a moment passes:)
#apm -vm
APM BIOS 1.2 (kernel driver 1.13)
On-line, battery charging: 88%
(reboot to r197:)
# apm -V
apm version 3.2.1
# apm -vm
APM BIOS 1.2 (kernel driver 1.13)
On-line, battery charging: 5%
(~30 minutes pass:)
... battery status critical: 5%
(~10 minutes more pass:)
... battery status high: 100%
(2 minutes later, it's back to "critical: 5%". Occasionally I can catch it momentarily back up at "high: 100%.)
Regardless of the battery reading, if I remove the AC cable the Zaurus dies without leaving any trace in /var/log/dmesg or /var/log/messages.
On a separate C860 (Sharp ROM) using the same battery as above:
# apm -V
apm version 2.4
# apm -vm
APM BIOS 1.2 (kernel driver 1.14)
AC off-line, battery status high: 100%