Your older laptop battery probably wouldn't charge anymore because it was "dead". They usually last only 2-3 years, at most, regardless of whether they're constantly kept a full charge or not.
The laptop isn't constrained by battery power limits when it has external power so - as you've observed - it's able to run "better" with all hardware at full performances. It'll of course run harder and hotter when sustaining peak performance, which can impose a little more strain on the cooling systems. But this really won't kill the laptop prematurely - the battery is almost always the first item to age and die on any (not broken) laptop, but keeping the battery at/near full charge doesn't accelerate battery death.
The real killer of these batteries is letting them deplete "into the red". If kept in that condition (stored) for long periods the chemistry in these batteries permanently degrades. If forced to simultaneously charge and discharge at high rates (like when the battery is at/near no charge and the laptop is plugged in to keep it running) they produce a lot of heat which also degrades their chemistry. If you get in the habit of always charging your battery before it gets seriously depleted, and always letting them charge uninterrupted until they reach full capacity, then you might get a useful battery lifetime of around 5 years or more.
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