Voltage have great effect. Power is electrical current (amperes) multiplied by voltage (volts). For example 1 ampere at 5 volts is just 5 watts but same 1 ampere at 20 volts is 20 watts which is much higher. Different charging technologies have different approaches to send maximum power from charger to the phone but generally most use higher voltage to transfer power above 10 W because most usb cables do not support more than 2 to 3 A of current. Before charging the battery itself phone converts back the high voltage to low voltage suitable for the battery (usually between 3.5 and 4 volts). If you are looking at software programs like aida 64 or ampere - they read the current at this stage so this is why more current at that level means faster charging because the voltage is already reduced to what is needed by the battery. Generally you have something like this:
1. Phone and charger negotiate needed power and they way it will be achieved based on the capabilities of the charger, phone and cable connecting them. Let's say that they agree for 20W which is 10 V with 2 A current.
2. Charger transforms 220 V AC to 10 V DC and phone starts drawing 2A of power.
3. Internally in the phone voltage is transformed from 10 V to 4 V which causes current to rise to 5 A which are fed to the battery.
Here is a great analogy showing the relation between voltage, current, power and resistance using something as simple as water hose.