Yes, and I will connect that supply to the Vin port on the board, right?
No. If at all humanely possible do not ever use "Vin" or the "barrel jack". Supply 5 V, reasonably regulated, and supply it to the "5V" pin. That is what the microcontroller requires.
I get confused reading stuff like this:
"The power source you connect to the Vin pin has to be 7 to 12 volts for the regulator to work reliably."
And that is sad. No wonder you get confused.
A very real danger is that the obsolete tutorials on the Arduino site and others misleadingly imply that the largely ornamental "barrel jack" and "Vin" connections to the on-board regulator allow a usable source of 5 V power. This is absolutely not the case. It is essentially only for demonstration use of the bare board back in the very beginning of the Arduino project when "9V" transformer-rectifier-capacitor power packs were common and this was a practical way to power a lone Arduino board for initial demonstration purposes. And even then it was limited because an unloaded 9 V transformer-rectifier-capacitor supply would generally provide over 12 V which the regulator could barely handle.
If you are asking this question, it is highly likely that you will wish to connect something else. In which case, the answer is regulated 5 V.
This is because the on-board regulator is essentially capable of powering only the microcontroller itself and no more than a couple of indicator LEDs. The on-board regulator might be able to power a few other things if it had a heatsink, but on the (older) Arduinos, it does not.
Powering via the "barrel jack" or "Vin" connections is asking for trouble. The "5V" pin is not by any means an output pin, if anything a "reference" pin but most certainly the preferred pin to which to supply a regulated 5 V.