I was told that I need to sync the new battery to the laptop.
1) I never heard of syncing a battery.
It's ancient, and was never called "syncing" to my understanding. With older batteries the architecture was different, and it was possible for the computer to think the charge left was different then what the battery had. (I believe it operated on a scale where the battery said how much was being used but never actually reported the charge remaining, so the computer had to estimate based on a value that it thought the battery was at). This happens with an old laptop if you leave it for a while- the battery continues to drain and when you use it again a few weeks later the computer still thinks the charge is the same as when you shut it off. The method for remedying this was to completely discharge the battery, and then recharge from 0% (some battery applets even provided this function).
However, this has not been an issue for laptop batteries since the early 90's.