I think that the “exchange” term involves the effects of echanging particles, i.e, expressing the statistics of electrons and the asymetry of the wave function. that is something you dont have in the classical problem and (I think) that is why you need a different definition for the quantum version of the correlation.
If you look at the hartree wavefunction definition , it is only a product, which can not (as far as I can see) make for the asymetry with respect to particle interchange.
Correlation in the other hand, is well explained in the classic picture, and is related to the many body nature of the
problem.
Also, the exchange is treated exactly by hartree fock wave function, and correlation (I understood) is treated exactly in post-hartree-fock (computationally prohivitive for large systems), but those terms are solved approximately in real dft.that is why you need to improve the level of theory in those items.
