Difference between the natural response and transient response of an RLC circuit
I know this is a late answer, but I've just come across this question and wanted to add my point of view.

Suppose we have a generic dynamic (and linear) circuit with one or more external input sources and with some initial conditions at the dynamic elements. That implies that we have chosen an arbitrary time instant as our "initial" time. We don't have to know what happened to circuit before that point, because all gets "summarized" into the initial conditions.

Then, thanks to superposition theorem, the total time response \$r(t)\$ can be decomposed as a sum of the response to only the external inputs (the so-called forced response) and the response to only the initial conditions (the free, or natural, response).

If the circuit is stable, the natural response will decay to zero after some time, leaving only the forced response, which will stand forever (provided that we don't turn off the input sources and they don't decay to zero too).

So, for a while starting at initial time, we will have a response composed of natural + forced response. This is called the transient response. But after this while the natural response will have gone to zero and only the forced response will remain. This is called the steady state response.

The steady state response will last until we turn off the input sources. That will leave probably the circuit "energized" with some voltages and currents into capacitors and inductors, which will generate a brief natural decaying response until the total energy of the circuit is dissipated and the output reaches zero. This last response is another transient, this time containing only natural/free response.

Summarizing: the transient response of a circuit is the response part where the natural response is still present, whether merged or not with forced response. So, it equals the natural response only if there's not forced response.