Charger 'A' seems to be doing things right to me, exactly how all mine do / have done and follows the logic of the charge cycle. Charger 'B' doesn't seem to have a soft start to current build up like 'A', and I'm surprised it ends the charge so quickly. Charger 'B' would be suspect in my mind, albeit, maybe something that can be changed in its settings. (As for selling your Dad your knackered old batteries......
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To add, you're best off always Balancing, as over or undercharging one of the cells could be causing your issue. It will also take longer when you do eventually balance them than if you do it every time.