As John says, the key is you have fixed-voltage (12 or 3.3) so a 5 cent resistor is the way to go vs. 44 cents each for the CL2 at Mouser (I couldn’t find the CL2 at DigiKey). The CL2 is at its best when the available voltage varies such as Dale’s example of passenger cars in conventional.
If you have a meter handy, measure the DC voltage across the 1 ohm resistor. This will tell you the current into the LED. 1 ohm makes the math easy – the voltage in Volts is the current in Amps. So if you measure 0.02 V across the resistor, then the LED is seeing 0.02 A (or 20 milliamps). If you measure anything more than 0.025V (25 mA) I suggest increasing the resistance.
As the guys suggest, for your house, I’d go with 12V and 3 LEDs in series plus a resistor.
The CL2 requires about 3V across it to deliver the 20 mA. So you can’t use a CL2 with your 3.3V bus to drive a white LED or really any LED for that matter. By the time the CL2 takes its 3V “tax”, there’s only 0.3V left for the LED which means you’ll have something closer to a DED (dark-emitting-diode).
Now here’s the nerdy part. The CL2 in series with 3 white LEDs may not deliver the full 20 mA with a 12V bus. The following chart from the CL2 datasheet shows this voltage “tax”. To deliver 20 mA, the voltage across the CL2 must be about 3V or so – otherwise it supplies something less than 20 mA or nothing at all if the voltage is too low. At 20 mA, this leaves about 9V or only 3V per LED – cutting it a bit close. Of course the magic of the part is it continues to control (limit) the current to 20 mA over an extremely wide voltage range. To contrast, I inserted the behavior of a 150 ohm resistor. Current starts flowing at a lower voltage than the CL2 but takes off to the stars with no looking back!