Should I connect the Arduino Ground to the 12V battery - terminal?

Im wondering if its ok that just the Arduino ground is connected to the pull down resistor.

Or should the 12V battery - terminal be connected to the same ground?

Hmm, i don't know...
How are you powering that Arduino ?

Is that FET an isolated switch ?

Perhaps someone has a remark about it in their sign off.

Either a 9v 6xAA battery pack or straight from the USB cable-laptop.

What do you mean "is the fet an isolated switch"?

So you are powering your Arduino from some other source than the car battery.
How does the FET know you are trying to control it ?
Would it be able to see your signal if it doesn't have a reference to compare it with ?
Such reference is GND.

So yes, you need to connect the battery 0 volts or minus to GND of the Arduino.

The FET mentioned by you isn't a relay or an optocoupler, is it (because that is not a FET) ?
Either of those would need the GND of the Arduino to be connected to its input (that would mean there are two input pins).

Marciokoko:
Or should the 12V battery - terminal be connected to the same ground?

Its vital. You may have caused damage to the MOSFET gate oxide by not doing so, they are static
sensitive and only tolerate 20 or 30V between gate and source before failing, and its easy to pick
up that sort of voltage floating about in the air near mains wiring (ie indoors).

Is your MOSFET logic-level? That circuit requires it to be so.

Gate-oxide failure can lead to partial working state BTW - test that the MOSFET is fully off when gate
is shorted to source directly.

I connected the grounds and it's working just fine. How could I test just the MOSFET to see if it's somehow damaged?

Marciokoko:
I connected the grounds and it's working just fine. How could I test just the MOSFET to see if it's somehow damaged?

If it was broken.... then you wouldn't have said working just fine. This means..... you already tested. It works... undamaged.

Ok well I thought the mosfet might get damaged in a way that current still Flores but maybe not the right amount. I guess I was wondering that if a mosfet was partially damaged it might let fewer amps through and that I could somehow measure the real vs expected amperage.