Note - a diode (or 2) should be added to the circuit, but more (or changing the dwg) later...
It's a "momentary" on push button - ie, it is on "momentarily" only whilst you push it.
So yes - as you described, spring loaded & stays out. It does not stay in or toggle (change over) a switch connection )ie, push-on, push-off, push -on etc)
The circuit requires the "12V source for pump" to be +12V. Without that, the relay will not latch on (ie, stay on).
I'm assuming that is with ignition on and the car ready to go etc.
With +12V available the "Push" switch, a momentary on with energise the relay - ie, energise the solenoid/coil and pull the upper contacts closed. (The push must be long enough to close the relay contacts - usually much less than 1 second.)
After the above, the +12V "Pump source" is conected through to the fuel pump AND also the relay coil, thereby keeping the relay energised and keeping the pump on.
The only way to open the relay to stop the pump is to either break that feedback connection (pump's +12V to the relay's coil), or remove the +12V "Pump source" - ie, either turn off the ignition, or the car's own fuel pump control has shut it off (hence you do not override the car's off control).
One problem - the momentary switch will supply power to the pump. This could burn out the switch etc.
Likewise, with te +12V Pump source & relay on, there is +12V supplied to the relay-side of the momentary switch. This is no problem unless the momentary switch's +12V is a different supply etc.
Those problems are fixed by inserting a diode in each side of the relay coil's top "T". I'll do that later....
Another problem - if your car is EFI, you can probably probably crank the engine with the fuel pump off.
That means the fuel injectors are dry whilst pulsing open & closed and this wears the injectors.
It can also introduce air into the injectors etc, but that should be blead out - it might just mean a bit of extra dry, or misbehavior for a while.
If it's a carby car, the engine will probably start and run until the fuel bowl runs dry.
This could place it in a freeway or road etc.
But otherwise it is a good occurrence because any would be thief can usually pick and engine or start kill system and can usually easily defeat it (most people have no idea how easy!), whereas a stall and no start after a short run could mean some problem so they leave it. (In the freeway, or on the railway crossing.)
A local anti-theft circuit many years ago used that principle - a simple timer would kill the ignition every 20 seconds or so until it was defeated. First stall - ok, it stalled; 2nd stall... hmmm; 3rd stall "To heck with this, this car is fudged!"
Anyhow, they are some considerations for HOW to "kill" a car - or not.