A 'double throw" relay will close one circuit when the coil is not energized. That circuit opens and another one closes when the coil is energized. Makes it an "either-or" switch function.
This can be done with a 12-Volt single-pole double-throw relay and two pushbutton switches. First, the 12 Volts connects to the relay's "common" pole, the one that selects either the NC when there's no coil current, or the NO that closes when the coil is energized.
One pushbutton type is normally open, and closes only when pressed. The other one is a normally-closed type, and only opens the circuit when you press it.
The normally-closed switch will go in line with the relay's coil.
The normally-open switch will go from 12 Volts to the hot side of the relay coil. The 12-Volts that comes out of the 'on when coil is energized" or NO (normally open) side WILL ALSO go to the hot side of the relay coil. It will hold itself in the energized condition until the normally-closed switch is pressed and cuts the coil circuit.
Can you read an electronic schematic diagram? I have discovered that not everyone does. If so, I can draw it up in that form.
And if not, I'll save myself the trouble.
But that's the plain-language version of what you described.
One output from the relay is hot when the coil is energized. The other output from the relay is hot only when the coil is NOT energized. One or the other is always on.
73