Natural, Forced and Total System Response - Time domain Analysis of DT
First of all, many textbooks use the following definition (but see this answer for a different one):

zero-state response (ZSR) = forced response

zero-input response (ZIR) = natural response

The zero-state response is the response of a system to an input signal given that the initial state of the system is zero. The zero-input response is the output of the system due to a non-zero initial state. For a discrete-time system, the initial state is given by the values stored in memory (i.e. in the delay elements). Since we're dealing with linear systems, the superposition principle can be applied, and the total response of the system is given by ZSR+ZIR. So if the initial state is zero, the total response equals the ZSR, and if there's no input signal, the output is given by the ZIR, which always decays if the system is stable.

In analog circuit theory you have exactly the same. There the initial state is the energy stored in capacitors and inductors. Your car engine analogy is less fortunate because it doesn't really work. You could come up with a mechanical analogy though as long as the system is linear.

It might be a good idea to formulate a new question concerning recursive systems (after having searched the internet and this site). But you're right that a system is recursive if not only the input but also past output values determine the current output. The reason why this is done is the fact that in this way certain system properties can be implemented much more efficiently than by only using the current and past input values. The problem with recursive systems is that they can be unstable, and that quantization effects (in signals and filter coefficients) can more easily result in numerical problems than for non-recursive systems.

PS: If you teach yourself I strongly recommend MIT's open courseware. You should probably start with 6.003 Signals and Systems.