My experience with Maxx battery:The part about the warranty restarting with the new one is not true. I have warrantied 5-6 walmart batteries over the year and EACH time they clearly told me the warranty on the replacement battery is only what's left on the original (bad) battery. You don't get a full warranty on the replacement - just what's left on the one you replaced.
In the three year free replacement period, you are correct. If it fails in two years, for the new battery, there is one year free replacement, 2 years prorated.
If you PAY for the battery (after 3 years, prorated), then it gets a 5yr (3replace,2 prorate).
I don't know about the cheaper batteries, but it probably works like the 3 year period above.
Lawn mower batteries are 90 days. Probably best to return and buy another if that is true.