Just want to make sure I understand this and have it correct.
The optically-isolated bit keeps the 5mA trigger input isolated from the 12v coil circuit. I'm assuming that this 5mA trigger is used in the scenario where you have a battery hooked to the 12v circuit +/-, and you have a separate, very small signal to close the relay. This would mean the 5mA trigger input needs to be 'tied' to high or low, which is why that product has resistors on it.
But I am using a switched 12v circuit, not a battery, so I don't need to concern myself with that. The switching of the circuit (i.e. the trigger voltage goes to 12v) will switch the relay open/closed instead.
Once the relay is closed, it's the same situation as the automotive relay (except the required trigger current), with a simple contact switch. And this is why, owing to the extraneous opto-isolation and accompanying resistors, you stated that it was
You have it more or less for what you need to do. Just to be precise...
There is no coil in the opto-isolator. The electro-mechanical automotive relays have a coil which pulls the contacts closed magnetically when trigger current is flowing through it. In the opto-isolator, a diode in the trigger circuit produces "light" which is detected by the relay circuit to switch on or off. This is why it is more efficient in current draw.
The isolation is between the trigger and the relay (the NC or NO) circuits. Typically you are using them for very different voltage and ampere levels or even AC in the relay circuit.
The 12VDC + and - is just to power the board since it has electronic components that need power to work. You set it to detect whether a separate trigger input connected to IN is going high (relative to the -) closes NO or going low closes NO based on the jumper. This is the common use.
In your case you can use the 12V trigger from the amp as both board power (draws about 40mA) and trigger input (draws about 5mA). So don't need a separate power source. When it is off, there is no power coming to the board at all and NC stays closed and NO stays open. When the amp powers the trigger, it powers the board via the 12vDC + and - and lets it detect the same 12V coming through the input as its trigger. Whether this is considered high or low depends on the polarity of how you connect the amp trigger wire as described earlier. You can experiment with both.
Yes, the relay side is the same as the automotive relay. Just a switch that closes or opens. In this case there are two switches (NO/common and NC/common) going in opposite directions of open and closed. So you can map the logic of what the low and high in the trigger is to the switch is closed or open one-one or inverted.