Why is the tachyon vertex operator $int d^2z :e^{ik.X(z,bar{z})}:$ integrated?

I understand that, from the state-operator correspondence, $$|0;prangle ;=;:e^{ip.X(0,0)}: |0rangle.$$
This is given in Polchinski, equation (2.8.9).
I am now trying to understand the S-matrix for $2times$tachyon $rightarrow$ $2times$tachyon scattering. My understanding is as follows.

We seek to calculate $langlepsi_f|psi_irangle$, where the initial and final wavefunctions are both two taychons. That is, we want to find
$$langle 0, q_1 ; 0, q_2 | 0,p_1;0,p_2rangle ; =; langle0|:e^{-iq_1.X}: :e^{-iq_2.X}: :e^{ip_1.X}: :e^{ip_2.X}:|0rangle$$
$$ = int DXDg ; e^{-S_{Poly} [X, g]} ;V[-q_1, 0]V[-q_2, 0]V[p_1, 0]V[p_2, 0],$$

where $V[p, z] = ; :e^{ip.X(z, bar{z})}:$.

However, this is not the result quoted in textbooks (e.g. Polchinski eqns 3.5.5, and 3.6.1). Instead the actual result is

$$int DXDg ; e^{-S_{Poly} [X, g]} ;int d^4z_i d^4bar{z}_i V[-q_1, z_1]V[-q_2, z_2]V[p_1, z_3]V[p_2, z_4],$$

citing diffeomorphism invariance. I see that my expression is not invariant under diffeomorphisms, whilst the second one is. However, why is the second expression the correct result? I see it in textbooks referred to a guess, but a guess of what exactly? Surely what we have now is not the overlap of four tachyons $<psi_f | psi_i>$ which we sought to calculate, but rather the overlap of some strange superposition of infinitely many tachyons. How does this relate to the $2times$ tachyon scattering amplitude at all?