Input impedance of inverting amplifier clarification
When we talk about the input resistance of a circuit, we're describing how it affect the other circuit that is providing the input signal.
Specifically, we want to know, in order to change the input current by i amps, how much do we need to change the input voltage? That's why the input resistance is, by definition, \$ \dfrac{\mathrm{d}v_i}{\mathrm{d}i_i}\$.
So what's the input resistance of this circuit?
The key point is that in this configuration, as long as we avoid saturating the op-amp output, the inverting input of the op-amp is a virtual ground. The feedback in the circuit operates to keep that node at 0 V. So whatever input current we want, by Ohm's Law, the required input voltage is \$\mathrm{R_{in}}\times{}i_{i}\$. Therefore the input resistance is Rin.
Then there are answers like this that mention that input impedance is infinity.
That answer was talking about the input resistance of the op-amp, which was assumed to be ideal, not the input resistance of the whole circuit.
The Photon's answer is absolutely right: ideal op-amp, etc, input impedance is Rin.
In a more general circuit, even one with non-linear components like transistors, input impedance is a small-signal (linearized), frequency-dependent quantity. It's important because it can tell the designer about loading effects between the output impedance of the prior stage and the input impedance of the next one. The general approach to calculate input impedance (or output impedance) is to inject a small current into the input node (di) and look at the resulting change in voltage of the input node (dv). Or, equivalently, to apply a small voltage (dv) and look at the resulting current (di) from your test voltage source. Then compute (dv/di). To make it completely clear that this is what's going on, take a look at my answer to How to compute input impedance, where I demonstrated how to use a circuit simulation program to compute and plot the input impedance by literally adding a test voltage source at the input and plotting a custom expression. Hopefully, seeing the voltage source V1 (or in your case Vin) drawn literally as a voltage source will make it clear how to go about setting up the calculations for doing it by hand!