Where does this formula for normal ordering in QFT come from?

Question

On this Wikipedia page you can find the following equation for free fields $$:\phi(x)\chi(y):=\phi(x)\chi(y)-\langle0| \phi(x)\chi(y)|0\rangle\tag{1}$$ But I don't understand where this comes from while it seems a very fundamental equation. I tried understanding it by switching the terms: $$\langle0| \phi(x)\chi(y)|0\rangle=\phi(x)\chi(y)-:\phi(x)\chi(y):\tag{2}$$ Then the following makes sense: you can expand $\phi,\chi$ in terms of creation/annihilation operators. The expectation value of a normal ordered product is zero so it makes sense that by subtracting it on the RHS the normal ordering term doesn't contribute. For example $$\langle0|a a^\dagger|0\rangle=a a^\dagger-:aa^\dagger:=(a^\dagger a+1)-a^\dagger a=1\tag{3}$$ What doesn't make sense is that $\phi,\chi$ are generally complex sums of creation/annihilation operators so after applying Wick's theorem I would expect there would be many more terms in the RHS of the second equation. So my question is why does this equation have this form? I feel like understanding this equation would really help my understanding of QFT so any help is appreciated.