In most books it is said that natural response is another name to zero-input response while in some resources is is mentioned that the classification is based on poles of transfer function and input.As the second definition is more theoretical then mathematical I'am having difficulty in finding natural response of certain systems where system function and input have same poles.
"Natural response" and "Zero input response" seem to been used as name for the output signal, when the input signal is zero, but inside the system there's some energy stored which causes something non-zero to output
An example: A heater in a room. Input=electricity. Output = How much warmer the room is when compared to the outdoor air. Let's have the input=0, let the room temperature (in centigrade) be = 20 degrees and outdoor temperature = 0 degrees. The room gets gradually colder as the heat escapes through the walls. The cooling curve vs. time is "Natural response" and "Zero input response". Its determined by the system state evolution law and the initial conditions.
I bet you remembered wrongly something when you wrote the last sentence. As already commented, it's nonsense.
If you have only the Laplace domain transfer function, you cannot get the natural response by any means without additional assumptions, because you do not know the structure of the system and how there's initially energy stored inside the system. You must have a (state variable) model which presents the energy storing subprocesses as integrators and how much they have charged content at t=0.
Of course you can find an equivalent state variable presentation for the s-domain transfer function and assume some initial state - not only for output, but also for every integrator inside the system. But that's not a truth of the modelled physical system, for ex. an electric circuit. To be truthful, the state variables must be the outputs of the real energy storing components ie. capacitor voltages and inductor currents.
The poles in linear system function: If you calculate the impulse response of the s-domain system function by taking the inverse Laplace transform, you get a sum of complex exponentials determined by the poles. Heaviside's general inverse Laplace transform formula for rational s-domain functions tells it explicitly. If the natural response for t=0 is seen as an extension of response to non-zero input before t=0, your last sentence has at least a couple of right words. The response after t=0 with zero input is a sum of the same complex exponentials which occur in the impulse response. A simple example is a filter with a resonant circuit. It oscillates with same frequencies which are included to the impulse response and they decay like in the impulse response. Their strengths depend on the state at t=0.
If you can simulate your system, let the input to be =0, but put some initial values to integrators. In a circuit simulator you must set the initial values for capacitor voltages and inductor currents. The zero input response is the result of the simulation. Delay lines make the case complex. You must define the signals which fill the lines at t=0.