I really don't have any words of advise for LiIon batteries because they don't seem to have the same pickiness about charging the other two had.
I will say don't expect the $40 battery to last as long as your original. You might get lucky, but every time I've gotten a cheap battery off of eBay it wasn't even worth what I paid for it. I have bought good batteries for 60-70% of the OEM retail price, but when you get below half price they generally don't have the life claimed or last the number of recharge cycles.
It is Li-Ion I wasn’t sure they even made laptop batteries in NIMH or NI-CAD.
Well if it fails within a year their warranty will cover a replacement, which is kinda why I want to get the most use out of it I can out of this year. Hopefully, running on batteries at least 4-5 hours a day, while being charged another 4-5 hours.