In the first picture the Arduino and the servo are both being powered from the same battery.
It is probably easier to power them from separate power supplies because they need different voltages. Just make sure the GNDs of the two power supplies are connected.
To power the Arduino through its power jack it needs between 7 and 12v and 7v power supplies are not very common. However a 9v power supply is too much for the servo. You can power the servo with 6v or even 5v) if you don't need the extra performance that 7.4v would give.
Another option is to power both the Arduino and the servo from a pack of 3 xAA cells (4.5v) or a regulated 5v power supply. In both of these cases the power supply should be connected to the Arduino 5v pin. However, be warned, connecting more than 5v to the Arduino 5v pin will damage the Arduino.
...R
PS ... the PP3 style 9v battery in the pictures is quite unsuitable. It cannot provide enough current. If you want a 9v battery use a pack of 6 x AA cells.