When I studied the resonant frequency of an RLC circuit in series and parallel, there was the definition that I read in books that the imaginary part of the total impedance of the circuit has to be equal to zero and the frequency where that happens is the resonant frequency. I'm fine with that, but there can be multiple transfer functions in a circuit depending on what output signal you choose, so for every transfer function there is 1 or more frequency values that generates the maximum value (amplitude) of the function. So is there 1 (or more) resonant frequency for every specific transfer function? Or are those values not resonant frequencies?
Yes, the resonants frequencies of a circuit made of whatever bunch of RLC are intrinsic to the circuit.
But there are two subtelties :
The stimulus and the measurment should not change the circuit. If the stimulus is a low impedance source (a voltage generator) then you can only apply it by insertion in a wire of the circuit you have cut. If it is a hight impedance source (current generator), then apply it simply to a pair of existent nodes. In the special case of a serie LC, you measure the resonant frequency by applying a current between the extremeties, but it turns out that can also find the same frequency by applying a voltage source and measure the current : This is a specificity to the LC circuit alone. If you do that with a more complex circuit, then consider that you have changed the circuit : it isn't the same circuit !
For a more complex circuit, you have N several resonant frequencies. If you make a measure somewhere at a point P, it can happen that the contribution of a resonance is 0 at this point P (mathematicaly, one says the residu of the complex function pole is 0). Take for example a circuit made of two independent LC (I mean non coupled). In real life this is not a problem because it's evident.