Does "rico" have a more specific reference to food that just "delicious"?
"Rico" in reference to food means delicious, but does "rico" it have the same specific reference that rich in English has, i.e., lots of fat or shortening?
2 Answers
- (of food) having a large proportion of flavoursome or fatty ingredients, such as spices, butter, or cream
- adj. Gustoso, sabroso, agradable.
Titles like this would seem to say no.
Comida rica y sana
Also if you look at recetas para comidas ricas they are not desserts, fried foods or things that you would expect for "rich food" in English. The spicy context (flavorsome) may apply, however.
They do you "rich in" with food as English does. (As Gekkosan stated)

Not a native so take it with a grain of salt.
Yes. In Spanish it can be a bit confusing, because technically, you can say:
Es un plato sumamente rico en grasas y condimentos ( a very fat and condiment-rich dish).
However, if you just say: "Es un plato muy rico"; everyone will understand that you mean "it's very tasty".
In other contexts, "rico" means simply "rich". Un hombre rico: a rich man. Un ambiente rico y fértil: a rich and fertile environment. Una chica rica: a rich girl, with a twist of double entendre.