The fact that $\phi(x)$ has the time in the operator identifies the formula as one in the Heisenberg picture. In the Schroedinger picture, operators are time independent, unless time occurs explicitly as parameter in the interaction.

Even outside of field theory, time correlations are awkward to represent in the Schroedinger picture, where only a single time is natural. One can put one of the times into the Schroedinger state, and the time difference remains in the operator expression, as a conjugation by exponentiated Hamiltonians.

Moreover, the meaning as a time correlation is apparent only in the Heisenberg picture in the Schroedinger picture it is just an obscure formal expression without ready interpretation. Thus anything using multiple times is natural only in the Heisenberg picture (although formally one can rewrite it also in the Schroedinger picture).

This shows that the Heisenberg picture is the fundamental one, and the Schroedinger picture is derived.