quantum mechanics - How exactly is "normal-ordering an operator" defined?

(In this question, I'm only talking about the second-quantization version of normal ordering, not the CFT version.)

Most sources (e.g. Wikipedia) very quickly define normal-ordering as "reordering all the ladder operators so that all of the creation operators are to the left of all of the annihilation operators." This definition is extremely vague, and I want to make sure I understand the actual definition.

If I understand correctly, people use the phrase "normal-order an operator" to mean two inequivalent things. Sometimes they mean "use the (anti)commutation relations to rewrite the operator so that it is normal-ordered (without changing the operator itself)." Under this (unambiguous) definition, we have that the normal-ordered form of the operator $a a^\dagger$ is $a^\dagger a + 1$. We can use this definition to put any operator into canonical form (up to a sign, in the fermionic case. We can fix this sign ambiguity by specifying a canonical ordering of the single-site Hilbert spaces.)

But sometimes the verb "normal-order" is used in a different way, which can actually change the operator. I believe that this definition is the one usually represented by surrounding the operator with colons. If I understand correctly, this procedure is defined as "use the (anti)commutation relations $\left[ a_i, a_j \right]_\pm = \left[ a_i^\dagger, a_j^\dagger \right]_\pm = 0$ to move all the creation operators to the left of all the annihilation operators, while ignoring the $\left[ a_i, a_j^\dagger \right]_\pm = \delta_{ij}$ (anti)commutation relation and pretending that its RHS were zero."

This procedure obviously seems a bit arbitrary and unmotivated. Moreover, it doesn't seem entirely well-defined. It's fine for products of ladder operators, but the problem is that under this definition, normal-ordering does not distribute over addition:

$${:} a^\dagger a{:} \ =\ a^\dagger a\ =\ a a^\dagger - 1\ $$ but $${:} a a^\dagger{:} -1\ = a^\dagger a - 1.$$

It's therefore not clear how to define normal-ordering for a general operator, i.e. a general linear combination of products of ladder operators. And of course, whether or not an operator is a nontrivial sum of products of ladder operators depends on how you write it; we can equivalently write the same operator as $a^\dagger a$ (only one summand) or as $a a^\dagger - 1$ (multiple summands).

From this, I conclude that (under the second definition) "normal-ordering an operator" is actually an abuse of terminology; we can only meaningfully normal-order certain particular expressions for some operators. Is this correct? If not, how does one define the normal-ordering of a linear combination of products of ladder operators?

Answer

The main point is that the normal ordering procedure $:~:$ does not take operators to operators, but symbols/functions to operators. This important point resolves various paradoxes created by abuse of language. For a full explanation, see e.g. this Phys.SE post and links therein.