Jobi-Wan:
I hope this is not considered thread hijacking but could you elaborate on this a bit?
With pleasure! 
Jobi-Wan:
So if the contrast is set by adjusting the resistance from pin 3 to ground, not by the voltage you set to pin 3, does that mean that pin 3 is a current source? And doesn't that also mean that the total resistance of the potentiometer matters more?
Pin 3 is indeed a current source. It is a "ladder" of five 2k2 resistors (R1 to R5 on the board) connected to Vcc and setting the contrast voltages for the liquid crystal. As follows:
There was a major mistake early on in the application of the HD44780 LCD driver chip where a designer copied something from a test circuit in the chip documentation where a potentiometer was used to set the contrast voltage. In fact, I have not been able to locate this misleading circuit, it is not in the present datasheet to hand which shows only the correct configuration (in figure 21, page 37) of a variable resistor in series with the negative supply to the contrast ladder. You may note that the negative 5 V supply is only required for the uncommon "extended temperature" version of LED displays, for almost all readily available, the contrast voltage is slightly less that 5 V and the resistor connects to ground.
Nevertheless, once made, this blunder became ingrained in almost all following designs and "instructional" literature, since few designers actually understood the magical workings of a multi-level multiplexed LCD - or cared to spend the time to do so.
The popular modules - the "1602", the "2004" and other variants - have the resistor ladder shown in figure 21, consisting of five 2k2 ("222") resistors "R1" to "R5", totalling 11k. R6 or "RF" is the clock oscillator of the HD44780 while R7 is 0 in the 5 V versions. R8, usually "101" as 100 Ohms is the LED resistor. The usual optimum contrast voltage - the voltage on the ladder between Vdd and Vo is between 4.5 and 4.8 V, corresponding to between 0.2 and 0.5 V on Vo. This then is most readily set by a resistance between 200 Ohms and 1.2k - if the supply voltage is actually 5 V - as this resistance becomes the correct proportion of the resistor ladder.
With the incorrectly wired 10k potentiometer, this resistance would be set with the potentiometer in the first (ground) tenth of its range (so ten turn potentiometers are often used) however the additional part of the 10k connected to 5 V draws an additional and completely undesirable 500 µA or so requiring the potentiometer to be set within the first twentieth of its range.
This is an absurd situation; if you must use a 10k pot, then it should never be connected to 5 V; the "free" end may be connected to the wiper or even to the ground end as this actually reduces the value of the potentiometer and makes contrast setting easier.
The correct value of the potentiometer, wired as a variable resistor with wiper connected to one end is 1k or 2k. This spreads the usable contrast setting to the whole range of the potentiometer.