Just to clarify, you mean that I should keep one of the terminals of the potentiometer disconnected. I should connect the middle terminal to the VEE and one of the remaining terminals to GND. No connections should be made to the 5V Line from the potentiometer, right?
That's right. Way back in the mists of digital time, someone copied the wrong circuit from the HD44780 datasheet, one which was only intended to show a parameter test circuit, and just about everyone including a number of "professional" designers has just mindlessly copied from them verbatim.
The incorrect connection needlessly wastes current, but also makes it more difficult to adjust the contrast. Connecting the potentiometer as just a variable resistor is an improvement, and using a 1k potentiometer instead as a variable resistor would spread the contrast setting over the whole range. Not that it is of course, that critical.
If you are using a 10k, connecting both ends to ground actually makes contrast setting easier.
Also, if resistor "R8" on the LCD module is "101" or 100 Ohms (very occasionally it is R7 or R9), that is the current defining resistor for the backlight and you do not need a series 220 Ohm resistor - the inclusion of which will just dim it slightly (and save current).