In Zavialov's book "Renormalized Quantum Field Theory" he defines the Chronological Product for arbitrary operators as follows (I-47): Given two operators in the form $$A(\phi) = \sum_{n=0}^\infty \frac{1}{n!}\int A_n(x_1, \ldots, x_n) :\phi(x_1)\cdots\phi(x_n):\mathrm{d}x_1\cdots\mathrm{d}x_n$$ $$B(\phi) = \sum_{n=0}^\infty \frac{1}{n!}\int B_n(x_1, \ldots, x_n) :\phi(x_1)\cdots\phi(x_n):\mathrm{d}x_1\cdots\mathrm{d}x_n$$ Their Chronological Product is defined by: $$T\{A(\phi)B(\phi)\} = \exp{\bigg\{-i\int \frac{\delta}{\delta \phi_1(x)}D^c(x-y)\frac{\delta}{\delta \phi_2(y)}\mathrm{d}x\mathrm{d}y\bigg\}}A(\phi_1)B(\phi_2)\Bigg|_{\phi_1=\phi_2=\phi}\tag{I-47}$$ with $D^c$ the causal Green function of the field $\phi$.
Choosing $A(\phi) = \phi(x)$ and $B(\phi)=\phi(y)$ (which corresponds to choosing $A_1=\delta(x-x_1)$ and similar for $B_1$) this corresponds to: $$T\{A(\phi)B(\phi)\} = :\phi(x)\phi(y): -iD^c(x-y)$$ which is indeed the Wick Theorem for two fields. But if now I consider the operators $$A(\phi) = -iD^-(x-y) + :\phi(x)\phi(y): \quad\Longleftarrow\quad A_0 = -iD^-(x-y), \quad A_2(x_1,x_2) = [\delta(x-x_1)\delta(y-x_2)+\delta(x-x_2)\delta(y-x_1)]$$ $$B(\phi) = 1$$ With $D^-(x-y) = i\langle0|\phi(x)\phi(y)|0\rangle$. Then the formula immediately tells us that $$T\{A(\phi)B(\phi)\} = A(\phi) = :\phi(x)\phi(y):-iD^-(x-y)$$
But we also know that the two cases are related because $\phi(x)\phi(y)=:\phi(x)\phi(y):-iD^-(x-y)$ so the Chronological order should be the same for both cases. But the two cases are not equal, because in one case we obtain the correct factor $D^c(x-y)$ while in the other case we obtain the factor $D^-(x-y)$.
Is there an error in my logic? Or this definition is therefore not coherent?
