Hi,
Yes i agree this is a good question.
There is some mystery to this when you first look at it because they seem to be quoting this and that and it seems to contradict in many places.
The relation of the gate voltage to the output current and resistance is not an exact specification. It varies between devices of the same part number and is affected by temperature. That means you could never design a circuit that takes an exact input voltage and passes a given current and it stays like that forever after. What you can do however is make sure your drive is beyond anything that could ever be needed, and that helps to ensure that you get the output you need.
For most MOSFET's the input range is from 0 to 20 volts. The 'turn on' voltage is somewhere around 4.5 volts, and that's where a typical MOSFET starts to turn on. It does not mean that every MOSFET you buy and hook up and test will turn on at 4.5 volts however, because it is just a characteristic specification not an absolute specification. It's characteristic because there are other types of lower voltage MOSFET's that will start to turn on at 1.5 volts, and this type is usually referred to as a 'logic' level MOSFET. That type will be fully turned on at 5v input or even less, while the 'normal' MOSFET might require 12v to be fully turned on.
The temperature has a lot to do with it too. If you want to make sure the MOSFET is fully turned on you have to feed it a high enough gate voltage to ENSURE that any specification of the input is met. If it starts to turn on between 3v and 5v, then that just means that the resistance at the output STARTS to decrease, so you have to feed it a much higher input to get it to turn on fully. 12v will usually turn it on fully but check the data sheet. You should be able to find a spec that tells you that the resistance will low at some input level, and going over that level means you should be getting the full turn on.
The logic level MOSFET's are used in lower voltage circuits where the drive signals can not be as high. These come in very low Rds types too.