Do I nedd a (zener) diode?
Yes otherwise you can damage the input of the Arduino. As it says use a 5V1 zener or if you don't have one of that value anything down to 3V3. As I said before :- Failing that a normal signal diode from the sensor to +5v should be used as well as that series resistor.
It seems that I get more range if I use the resistor. Is this possible or is it just my imagination?
What resistor? If it's the series one then you must use it to allow the zener diode to work. If it's the pull down one then it drains some of the current from the input pin allowing the voltage you generate to give a bigger input. You could also try a 0.1uF capacitor across the input to hold the voltage spike up a bit more and offer a bit more protection against over voltage from the sensor.