Linux – How to detect the presence of a battery

I don't know why you're looking for corroborating evidence from the system, when you have solid empirical evidence. The battery won't hold a charge. Period, end of story.

I'd be happier if I saw a high charge cycle count. Lithium ion isn't good for more than about 500 charge cycles.

Another datum, which won't appear in the places you've been looking, is how much of the laptop's usage life on this new battery has been while plugged in, and whether it's frequently gone through a full discharge/charge cycle.

Too many people use laptops as if they were small desktops, leaving them plugged in most of the time. This is not healthy for the battery. The analogy I like to use is that a toy balloon will last longer if you blow it up only part way, rather than blow it up as far as it can go and leave it there. A full charge on the battery increases some of the stresses on it.

There's (out-of-tree) module called tp_smapi, which provides access to (amongst others) access to the battery-related functions of the embedded controller. This allows you to do things like setting the start/stop charging thresholds, charge-inhibition timeout and also force discharge of a battery. Most distributions have a tp_smapi package, providing the module, otherwise you could still download the sources from Github and build them by hand.

When loading this module, it'll provide you with a sysfs interface under /sys/devices/platform/smapi/, one directory for every (possible) battery called BATn (where n would be 0 or 1 in your case) and some files you could write to. The file that could be the solution to your problem is called force_discharge. By writing 1 to it, you'll tell the embedded controller to forcibly discharge the according battery (this even works on AC, which allows you to »recalibrate« the battery as is possible with the Thinkpad Windows-Tools) — 0 disables forced discharge, accordingly.

I'm a bit puzzled that your internal battery is used first, though. I had a X61s with the additional battery-pack and AFAIR it used the external one first (which is… intelligent, since at least the X61s e.g. didn't use the external battery for suspend-to-RAM for obvious reasons, where it would be bad to have the internal battery discharged to zero). Hrm.