If you really want to understand the naming of animal groups, it might help to classify the types or origins of classification terms. There are words that have simply been borrowed from the past (e.g. an exaltation of larks) or from a particular language, dialect or regional speech. These are often popular among poets.
There are other words that are more commonly used. For example, most people would describe a group of larks as a "flock," not an "exaltation."
Scientists may use still other words.
Some of the most common collective words that I'm aware of are "herd" (generally applied to ungulates, or hoofed mammals), "flock" (of birds) and "school" (generally applied to aquatic animals, especially fishes).
P.S. The term "flock" can indeed be applied to geese and crows.