Chemistry - What kind of 'product' is the 'product' in 'ionic product of water'?
Solution 1:
Per Zhe's comment, in the specific context of the term ionic product of water, the word product is used in the sense of the result of the mathematical operation of multiplication:
The ionic product of water equals the concentration of $\ce{H+}$ times the concentration of $\ce{OH-}$:
$$K_\mathrm w = \ce{[H+][OH-]}$$
(Strictly, per MaxReid's comment, the ionic product water is of course the product of the activities of $\ce{H+}$ and $\ce{OH-}$, but concentrations are commonly used in routine/non-rigorous calculations as they provide good accuracy for dilute solutions.)
This sense of the word product applies any time one is referencing the mathematical description of the equilibrium among the ionic forms of a given species. This usage also includes the solubility product (again with the above caveat about activities):
The solubility product of silver chloride equals the concentration of $\ce{Ag+}$ times the concentration of $\ce{Cl-}$:
$$K_\mathrm{sp} = \ce{[Ag+][Cl-]}$$
As an aside, note that the main exception to this terminology is when the expression describing the distribution of species requires division by one or more quantities. In this case, the term to use is quotient, for which I expect little ambiguity exists:
The reaction quotient of the reaction
$$\ce{A + B->C + D}$$ is
$$ Q = {\ce{[C][D]}\over \ce{[A][B]}}$$
However, to be complete: chemistry does also use the word product in the other sense you describe, as in something being produced by a reaction. Due to this, the word product in the phrase the ionic product(s) of X could also bear this meaning. For example:
The ionic products of the aqueous decomposition of dinitrogen tetroxide at neutral $\mathrm{pH}$ are nitrite and nitrate ion:
$$\ce{N2O4(aq) + 2H2O -> NO2-(aq) + NO3-(aq) + 2H3O+}$$
In general, I think the difference boils down to whether one is talking about the ionic distribution of a specific species of interest, as opposed to the transformation that occurs in a particular reaction of interest.
As I noted above, though: In the specific case you are asking about, product is used in the sense of the result obtained from multiplication.
Solution 2:
The product $K_w$ is a mathematical construct.
The ion species produced from ionization of water are not mathematical constructs. They are resulting entities (actual, tangible "things").
The word does have two meanings, but the essence in both cases is "result". There isn't really a case when both meanings would apply to the same "result".