1 : tending to revive, cheer, or invigorate
2 a : sincerely or deeply felt b : warmly and genially affable
M-W neglected to mention that CORDIAL also refers to a candy in which a fruit filling is placed in a chocolate shell. This is most common in the genus: cherry CORDIAL. (The above image strikes me as both violent and sexual) I did locate a more exotic huckleberry CORDIAL made by the Farr Candy Company ("Quality since 1911"). I also found other CORDIALS that break the traditional boundaries: the rum CORDIAL, as well as the vacation CORDIAL mix, which includes pina colada, strawberry, daiquiri, and mojito candies.
CORDIAL also refers to a popular UK drink more commonly known as Squash, which is a fruit concentrate mixed with water or seltzer. A glorified Italian soda? We used to make this horrible drink at the coffee bar at Whole Foods called "Dr. Smoothie," which was almost exactly this. Perhaps it would have sold better under the name "Squash."
The word CORDIAL also describes any number of sweet liquors flavored with fruit, as well as many sweet fruit-based cocktails, most of which use said liquors. Here are some varieties I discovered:
As to what any of this has to do with sincerity or genial behavior, I have yet to determine.