But "fully on" also depends on the current.

Depends upon the definition of "fully on".
For the minimum ON resistance, which is basically independent of current, you need to use the Vgs as shown in post #4.
See the voltage drop at near zero current in Fig 2, in post #2.
The on-resistance varies with Vgs, even at low currents

 

The datasheet for the IRL540 says that the maximum allowed Vgs is 10V and it is guaranteed to conduct extremely well (its on-resistance) with a Vgs of 5V so use a Vgs of 5V.
It shows a guaranteed higher on-resistance with a Vgs of 4V then you can calculate if the voltage drop and heating are low enough.