I’m going to assume you’ve already seen Should I remove the battery if I leave my laptop plugged in?
What’s interesting here is that yours is a different — almost opposite — situation: infrequent use.
That actually makes this more difficult. Let’s review the options.
Option 1: leave the battery in and plugged in
This might be the easiest option to rule out.
By leaving the laptop battery in and plugged in, it’ll be in a state of constant charge. This is less than ideal for most batteries, and will shorten their life. Ironically, by keeping them continuously fully charged, they lose their maximum capacity slowly over time and discharge more quickly.
If you were using the computer — for example, if you left it on 24 hours a day — this might make some sense. The battery will still degrade, but at least you’re getting some use out of it while it does so. Coincidentally, this is what I do for my two laptops that have been relegated to “on the shelf, but always on” roles in my own computer menagerie.
But if the machine’s not even on, I can’t make an argument for this as an appropriate solution.
Option 2: leave the battery in, but not plugged in
This might be a reasonable approach, depending on how often you plan to fire the machine up.
The battery will slowly discharge, of course, while the machine is off. But if you turn it on often enough, it’ll work well. Unfortunately, I can’t tell you how often “often enough” really is. I will say that if the battery drains to the point of not being able to boot the computer without being plugged in, it’s probably been too long for this approach to make sense.
And yes, once you do turn it on, make sure to plug it in and leave it plugged in long enough to let it charge fully.
Option 3: remove the battery
If you don’t plan to use the computer for “a long while” (and as before, I can’t tell you how long “a long while” might be), then removing the battery might well make sense. The battery will still discharge completely, but it’ll probably do so more slowly than if you’d left it in the machine.
And once again, when you do re-insert the battery and turn on the machine, I think it makes sense to let the battery charge fully before turning things off and disconnecting.
Option 4: do what’s convenient
There’s a lot of variability here, depending on exactly what laptop you have, what battery you have, how good it was to begin with, and how often you find yourself using the machine.
In the long run, I’m not sure any of the alternatives make a significant difference. The bottom line is your machine is old enough that you’ve elected to move it to your second tier, so it’s safe to assume things aren’t pristine. The differences between the options above — particularly between options two and three — probably aren’t enough to warrant going too far out of your way.
Do what’s convenient.
It’s what I do. And what’s convenient for me is to leave the batteries in their respective machines, whether or not the machine is in use, or whether or not the machine is plugged in. My experience is that by the time a different approach might make a noticeable difference, the machine is so old as to be unusable for a variety of other reasons.