Arduino Wall Wart and Relay Board Power Supply
You will need the relay board.  The Arduino output 5V and so you can't use it to directly power the 12V half of the circuit.  

If you can split the power from the wall wart so it powers the 12V circuit and the barrel connector on the Arduino then that should work.  If you can, then you will need to ensure that the 0v of the Arduino and the 12v circuit are tied together, otherwise the two halves of the circuit will be using different versions of 0V and you will get interesting results.

If you want to use the setup in your second picture then no you don't need the relay board, but that's only because the whole circuit is 5V.

"How do you split the power?" I hear you say, "do I need any extra hardware?"

That depends, how attached are you to your wall wart's cable?  The way I have done it is to UNPLUG the wall wart, cut the cable in "half", strip a cm of each wire and plug both ends of it into two empty rows on you bread board.  Then use jumper wires to take an additional feed off to the 12V circuit.



CAUTION Ensure you get the polarity of the wires the right way around.  You can test this with a volt metre.  I suggest red electrical tape to mark the +ve wire.

IF YOU ARE NOT CONFIDENT DON'T DO IT!