powering servo seperately

I am i understanding this correct? I was reading some forums about powering servos correctly. Lets say i have a 9v batterry.
I connect the signal to the arduino and servo. Check.
connect positive to battery and servo. check.
connect servo ground to arduino ground? What about the the ground on the battery? what does that get connected to?

I connect the signal to the arduino and servo. Check.

Connect what signal? A 9V battery (poor choice) does not generate a "signal".

The signal wire on the servo connects to the Arduino.
The positive terminal on the battery (when you get a real one) connects to the positive lead on the servo.
The ground terminal on the battery connects to the ground side of the servo AND the ground pin on the Arduino.

Most "hobby" servos run from 5 to six volts, not nine.

If you have a stable power supply such as a regulated 5V one or three alkaline cells (at least AA) for 4.5V, you can power both the Arduino via its Vcc pin and the servo. If you run the servo from 6V (four alkaline cells) then you need to figure out how to power the Arduino, either something more than 7V to the "Vin" or 4.5V on "Vcc".

In any case all grounds - Arduino, input device, servo, each battery - connect together.

Maybe this schematic can clarify.
The coloring is mostly:
ctrl: Yellow or white
+: red
GND: black or brown
Most servos exept Airtronics have the +lead in the middle.

typical servo wiring setup.