With My Little Pony: The Movie (2017) we were given this series first introduction to Sea Ponies, which were to be revealed that they were Hippogriffs in disguise. Now as I see it the examples we've seen through season 8 seem to suggest that these hippogriffs are pretty similar to what most historians describe them as in mythology. However I make a point to state the following: These are all "pure" Hippogriffs, yet what about those who were not born of a Hippogriff mother and father? (See example photo)

(Silverstream here has two hippogriff parents and a younger brother)

Now as far as my headcannon is concerned, what I perceive as a "natural" hippogryph is simply that: the offspring of a pony and a griffin.
Unlike "pure" hippogriffs, "natural" ones are primarily carnivorous. They will hunt for food if necessary and often look similar to this:

Further more, most often than not they are looked down upon in society (primarily for being of a mixed species).