What a load of pseudo-science was the answer. The Li-Ion chemistry has the same 3.7V rating since at least 15 years ago, with 4.2V cap voltage. I still have the dead battery of a 2003 Compaq laptop and the cells are 2200mAh (2P4S) with the same specs as today's cells when I found the Panasonic datasheets. The chemistry has advanced for all cells, they are more robust, have higher capacity, longer life, and for Li-polymer increased cap voltage to 4.3V and 4.4V, but those cells are pouch type and used in smartphones, tablets and the most advanced laptops (those metal body slim laptops) due to higher energy density. Regular laptops, with low specs, stick with the cylindrical type cells, the cheapest and also more robust. The cylindrical 18650 type cells will work fine with any BMS controller from years ago, because it was designed with the specs still valid today.
The only thing that will be different will be the time required to charge the new cells if the capacity is higher than the original, since the peak current is fixed from the factory. The charging algorithm is the same CC-CV, constant current until a certain voltage, then constant voltage at 4.15-4.2V, so higher capacity will require higher time to reach CV mode, but this is absolutely fine.
Your problem, by now probably gone, since this post is 2 years old, is the BMS controller that doesn't know the cells have been replaced, and uses the same data of the degraded cells. The procedure to reset/wipe controller data and start fresh, with new capacity data, is proprietary. The battery has an I2C bus that is accessed by the OS or BIOS. But there is hope, since the controller is continuously monitoring the battery pack for actual charge capacity, at least two full cycles charge-discharge should replace the old data. Of course, the controller will report the original capacity set at factory calibration, that is stored somewhere, probably in its non-volatile memory, if it has one, or in a separate non-volatile memory chip on the board. But if the measured capacity is the one of the new cells, then is fine and over few cycles will report correctly the charge state.
I just replaced the cells on a Vaio laptop (10.8V, 3500mAh, 2P3S) and initially it reported the 25% left capacity from the old cells, but after 2 cycles it went to ~5000mAh of the new cells. At the third cycle it reported correctly the charge state, at 2% left the laptop stopped, then reported a full charge 5200mAh. The cells are taken from two fresh HAMA external battery packs. What was interesting after first boot was the reporting of charge cycle count as zero, taking out the cells left the controller without power and erased the number of charge cycles, so it was volatile data. I don't know where the old measured capacity is stored, but now is ok at 10 cycles. Yeah, the controller still says the designed capacity is 3500mAh, but the measured one is 5200mAh. The laptop now works for about ~7 hours of browsing and movies, from the original ~4.5 hours. The cells cost was 25$, while a replacement battery from a third party supplier costs 60-70$. And the satisfaction of fixing this thing and reviving and old laptop that still works great, priceless.
