My question is, as the 2N2222 is a NPN transistor, can I not simply attach the 12V+ direct to the relay, and then use the NPN on the GND side of the relay, activated by the High Output from an Arduino pin ( using the built-in pull up resistor, and an in series resistor to reduce the pin output to the current required by the NPN ?
Yes (except that bit about the pull-up resistor), this is low-side switching as opposed to the p-channel high-side switching of the linked example.
However you will need a diode across the inductive load to prevent inductive spikes - have a look at other relay examples for how to do this.
You also need to check that for the relay in question the transistor is able to switch enough current, and choose a good value for the base resistor (large enough to avoid overloading the Arduino pin, small enough to drive the NPN into saturation). Don't rely on the internal pull-ups they are very weak (20k to 50k), drive the pin as an output.