The example is not correct.
A led needs a constant current.
So you can't feed a 12V led with a 12V battery. The current will be a lot too much or a lot too less.
I assume that there are specific power supplies for those 12V leds.
Normally a led with a resistor is switched on and off.
It's quite obvious that the LED's used in the example are LED's sold with pre-mounted resistors for 12V operation, that's why it says 12V LED's
The high power LED's OP has found are meany for Halogen light replacement, so they have built-in power regulation and are fine for 12V operation.
To control the LED's with the arduino, do like on this diagram:
The M is the load, place the LED-module here.
The mosfet can be replaced with a NPN transistor.