Different forms of the LSZ reduction formula

I'm studying Chapter 7 section 2 of Peskin and Schroeder on the LSZ reduction formula, on page 227 they write the LSZ reduction formula

$$\tag{7.42}\prod_1^n \int d^4x_i e^{ip_i\cdot x_i}\prod_1^m\int d^4y_j e^{-ik_j\cdot y_j}\langle \Omega|T\{\phi(x_1)...\phi(x_n)\phi(y_1)...\phi(y_m)\}|\Omega\rangle\thicksim_{p_i^0\rightarrow E_{p_i}, k_j^0\rightarrow E_{k_j}}\bigg(\prod_{i=1}^n\frac{\sqrt{Z}i}{p_i^2-m^2+i\epsilon}\bigg)\bigg(\prod_{j=1}^n\frac{\sqrt{Z}i}{k_j^2-m^2+i\epsilon}\bigg)\langle p_1...p_n|S|k_1...k_m\rangle.$$

Then Peskin and Schroeder explain that we "compute the appropriate Fourier transformed correlation function, look at the region of momentum space where the external particles are near the mass shell, and identify the coefficient of the multiparticle pole".

However wikipedia LSZ reduction formula simply says the two sides of 7.42 are equal. Greiner's field quantization book chapter 9 agrees with wikipedia. So which one is correct?

Both are correct, although in Wikipedia & Greiner it is implicitly implied that the inverse propagator factors multiplied with the correlation function constitute removable singularities, since the external momenta in the $$S$$-matrix are supposed to be on-shell.