The dominant effect in exciton coupling is usually the so-called Coulomb coupling contribution, that is the Coulomb interaction between transition densities of localized excitations. At short distances, Dexter-type exchange effects are discussed to play a role, which are not well described by (semi)local functionals in time-dependent density functional theory (TDDFT) calculations. Overall, a large effect of the percentage of exact exchange on the resulting exciton splittings is known. Subsystem TDDFT allows one to analyze the exciton coupling mechanism by distinguishing direct from indirect effects, that is, changes in the actual coupling mechanism from modifications in the underlying local excitations. Our analysis shows that the strong influence of exact exchange is not due to a direct Dexter-type (exchange) coupling, but rather to an increased Coulomb (or pseudo-exchange) coupling triggered by a change in transition densities. This is demonstrated in calculations for 2-pyridone and chlorophyll dimers. We finally propose a route to efficient calculations of excited states of large pigment aggregates with hybrid functionals, which so far has been out of reach for quantum chemical methods.