That is correct.
With no power applied, there is no path to power the relay coil. When the switch is pressed, power reaches the coil and the contacts close. From then on, until the power is removed, the contacts keep the power connected to the coil.
There is a simple modification you could consider, it isn't essential but it will increase the life of the relay. You connect a diode (example 1N4006 which costs pennies) directly across the relay coil with the cathode (banded end) to the positive side. Assuming your car has a negative chassis, the cathode would be at the top in your schematic. The purpose is to reduce the high voltage spike you get when the relay turns off.
I'll try to explain without getting too technical: when a current flows through the coil it produces a magnetic field, it is this field that pulls the contact mechanism to close them. When the power is removed, the magnetic field collapses and the process reverses, a voltage is produced by the coil. How much voltage depends on how fast the field collapses but it can be several hundred volts and importantly, the polarity is also reversed. So for example, if you connect +12V to one end of the relay (as you do by pressing the switch) then disconnect it, the +12V end of the coil might go to more than -100V. The spike is very short, maybe only one thousandth of a second but it happens at the time the contacts have only just started to open so the gap will be very small and spark will jump across it. The spark eventually erodes the contact surface and reduces the relay life expectancy. A diode only conducts in one direction so if you wire it in the normally non-conductive direction it does nothing but when the polarity reverses, it does conduct and safely 'shorts out' the spark voltage.
Brian.