"pigeon" and "dove" are both words for the same family of birds. The bird most people think of with the word "pigeon" is the rock dove (https://en.wikipedia.org/wiki/Rock_dove) or domesticated / feral variants of it.

Yep, but importantly "pigeon" and "dove" are not exactly interchangeable words, there is just no consensus for which is which.

https://en.wikipedia.org/wiki/Columbidae

Toads and frogs are another pair like this, where there is no clear distinction

https://en.wikipedia.org/wiki/Frog

True. But I did not ask about "pigeon" and "dove." I asked it about "pigeon" and "mourning dove" which are unambiguously different species. Different genuses, even. Zenaida macroura v. Columba livia.

Homing pigeon has a nice response :)