My previous linear regulator had a flaw, which has been rectified in this circuit. The flaw was that the reference voltage completely depended on the input voltage, due to which variations in the input voltage would cause variations in the reference voltage (and hence in the output voltage).
(See the problem here: http://everycircuit.com/circuit/6186936728551424)
Now the output of a linear regulator (or any other regulator as a matter of fact) never changes, even if the input voltage varies (duh). That is now the case with this regulator, which is pretty much a non-inverting amplifier. Variations in the input voltage don't affect the output voltage anymore (it does affect the output voltage by 0.07V in the case of this circuit because of huge input variations and because the reference voltage is coming from a zener, but that won't be an issue if a tl431 is used).
Adjust the potentiometer to adjust the output voltage (but adjust the input before doing that).