Problem with Wick's theorem (normal ordering of a contraction)

Question

Taking the example of two bosonic fields, Wick's theorem is \begin{equation} T\{\phi(x_1)\phi^\dagger(x_2)\} = N\{\phi\phi^\dagger\} + N\{(\phi\phi^\dagger)_c\} \end{equation} where the subscript $c$ denotes a contraction of the fields. Since $(\phi\phi^\dagger)_c$ is a commutator (more specifically propagator), this reduces to \begin{equation} T\{\phi(x_1)\phi^\dagger(x_2)\} = N\{\phi\phi^\dagger\} + (\phi\phi^\dagger)_c. \end{equation} My question is that since we know normal ordering of a commutator is zero, why don't we have $$T\{\phi(x_1)\phi^\dagger(x_2)\} = N\{\phi\phi^\dagger\}$$ only instead of the above? In QED, by expanding the $S$ operator, we also get many terms involving normal ordering of contractions between fields (and other uncontracted fields) but these are not zero. Why? Is the normal ordering of a commutator and contraction different from each other? What am I missing?