Open-circuit means that the input is effectively disconnected - a high resistance, meaning no current flows, and the input can 'float' to whatever voltage it likes.

Zero volts means that the input is connected directly to the 0 supply line with a low resistance, meaning that current will flow if necessary to keep it at 0V.

The relevance to the phone mute input is that if it's open-circuit, it may wander up and down at whim (depending on such things as static build-up behind your dash), spuriously triggering the mute. If you put a 10K resistor between the input and ground, then it can never be open-circuit (when it's disconnected, the resistor pulls it back to 0V).

If your device presents 0V and open-circuit as its two states, you need a resistor to +12V so that open-circuit appears as +12V (if you had resistor to 0V, you wouldn't be able to detect the two states).

Does that help?