Also, do some googling on battery calibration. Many older laptops used to suffer from battery level calibration issues and manufacturers produced a calibration routine to reset them. This was usually an issue with the laptop 'thinking' the battery was discharged before it actually was. The calibration usually effected the lower end of the battery metering rather than the top end.
A laptop I had several years ago required the user to fully charge the battery for X hours and then FULLY discharge it until the laptop shut itself down. A full recharge then reset the calibration of the batteries internal charge level management.
As has been stated, a NOS Lithium Ion battery can be a risky purchase as officially Lithium Ion batteries become OLD and spec not guaranteed at 5 years of age. Brand new Lithium Ion batteries , and others, require several charge/discharge cycles before they will produce their full capacity.... it's a chemistry thing !
A good test of a batteries performance is to download a battery test program (there are many free ones available) and see how long the battery lasts from full charge to auto shut-down. The programs simulate normal usage with Word etc so give an idea of total continuous run time.
I once tested my six Dell X300 batteries and the best one was NOS. It exceeded the manufacturers spec for run time and it was at least 8 years old but unused. The used batteries varied from around 90% capacity to less than 50% remaining. It was an interesting little test and the PC did all the testing work itself via the test program.
In my experience, you cannot increase the capacity of a tired Lithium Ion battery by charge/discharge cycling as the battery chemistry is basically worn out. A new or unused battery is a different case as I have already mentioned as you are effectively activating the chemistry within the cells that has been dormant for a long period and requires some activity to 'wake it up'.
Oh and before I forget...Compatable batteries are a bit like Russian Roulette... I have had some excellent compatable batteries for camcorders etc but have had bad experiences with some 'Uniross' digital camera batteries from China that I believe were probably fake. They had less than 50% of the capability of the OEM battery yet claimed greater Ah capacity ! Many compatable batteries are fine but some are overstated in terms of capacity and others have a relatively short operational life so what started out as a 3000mAh battery quickly degrades to 1000mAh due to chemistry degradation
Happy Easter Everyone