The charger adapter (the device that plugs into the wall), is only half of the charging system. The other half is internal to the phone and controls what the battery receives and how fast. It's the job of the charger adapter to convert the high 110 volt US (or 220 volts in Europe), alternating current house voltage, and from 15 to 20 amps (1650 to 2200 watts), into a much lower 5 volts direct current, and .5 to 2 amps of current, so the phone can then safely manage the charging of the battery.
Voltages output to the phone on the Micro USB plug end are specified by the phone's manufacturer to be within a range of typically a 10℅ tolerance, meaning if it's specified to be 5 volts +/- 10℅, it's able to be between 4.5 and 5.5 and will still perform within specifications by the manufacturer without doing damage to the battery or underperforming. If the voltage of the charger adapter (the device that plugs into the wall), is too high, it can definitely cause harm to the battery.
As for wattage (or current), as long as it's high enough, it will perform as well as the manufacturer's specifications. It could be ten times higher than the current specified but the charging system in the phone will only allow as much current to reach the battery as the manufacturers want them to get and the battery won't charge any faster. If it's lower the battery will simply charge slower.
So to summarize, voltage is very important and must be carefully controlled. Current on the other hand is not quite as critical. However if the phone's charging system, which is internal to the phone were to fail and allow more current than specified it would charge the battery faster and potentially too fast to prevent doing damage to the battery.
If one of your charging adapters is underperforming another then it's either putting out too low of a voltage or current and as a result the battery would charge slower, but this wouldn't damage the battery. In fact it would actually be easier in the battery and extend the life of the battery.
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