Why are different currents used for charging batteries with the same voltage output?
Batteries are usually charged with a bit more voltage than their "full" voltage, but the details vary widely.  Often batteries that are deeply discharged need to be brought up slowly until they are reasonably full.  Most batteries can be charged with simple logic that applies some fixed voltage or some fixed current, whichever is lower, if you're willing to give up some charging time.  Even that doesn't apply universally though, like with NiMH.  If you want the fastest possible charging time, you have to carefully do what the datasheet says to do.

A 2 A charger isn't going to produce a higher voltage than a 1 A charger, but it has more current capability.  That means it is intended to charge a larger battery, all else being equal.

For the best answer, read the datasheet for the particular battery you want to charge and don't assume it applies to any other battery.