If you supply the 5V on the Vin Pin, there's an on-board regulator that gives you a regulated +5V output on the 5V power pin of the arduino while powering the arduino itself.
This is not correct. While it is true there is an on-board regulator, it will not reliably work if given 5V in. The on-board regulator (like the 7805) really needs about 2V and so Vin should be at least 7V.
I've power brick which 5V output and it seems to be 'stabilized' will I need a further voltage regulation
Generally power bricks are regulated. However, they also tend to be switching power supplies which may have a minimum load requirement. In other words, they are regulated provided enough current is being drawn. One simple measurement is to measure its no-load voltage. If it is 5V you should be fine. If it goes higher than that, you might need to do more investigating.