Welcome.
In my case the screwdriver is marked PBSA 12 A1, the original battery is marked DCBK-70-12 12v 1.5Ah and the charger from this set is JLH201261500G.
Charger:

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Unfortunately I bought a battery in Lidl years later, but from the X12V team series with the designation PAPK 12 A3.
This is how they both look:

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I have the feathers in the new acku already cut down and will eventually be glued with two-component glue new, in the place where the old one has them.
However, the problem I have with the charger is that it gives 13.6v on its contacts, yet both LEDs give no indication. This problem started while the old battery was still in place, now that I look at it this voltage persists even after inserting the batteries, so I think the charger is charging, but I still need to check this carefully.
As for the old battery, there is no voltage between the -/+ leads on the T contact and internally it goes via the diode to the n-mosfet.
On the new A3 battery, on the other hand, there is almost the same voltage between "-" and "C-" as between "+" and "-", although "+" and "C-" are not internally connected, so there is additional electronics there.
As for the ID field, in this video the guy shows that there is a 24k resistor connected to the T field, so presumably this is a way for the new charger to handle the old batteries (7m26s):
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That's it from my side for the moment, and I'm getting on with finding out why this charger of mine is not signalling charging.
Edit:
Damage to the charger found, it turned out to be a broken path right next to the first green diode, which is shared by both diodes, so I think the set will last for many more years

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Finally, paste in the new interlocks:

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