I recently watched an episode of My Little Pony where Twilight says "Nobody". It's in the episode "Over A Barrel" where she says "Why won't anybody be rational or reasonable?" This runs contrary to the majority use of "Nopony". Was this an error, or are there specific rules for when they use these terms?
As now, it is somehow unclear if a rule does even exist officially.
Let me explain. From an external viewpoint, I am not aware of any official comment by the staff that would prove the existence of a written "style guide" that regulate the use of words like "....[pony]", using "hoof" instead of "hands" (or claws) and so on. This obviously doesn't mean such rules do not exist - I expect they do have a style book to adhere too. The problem is that we don't have access to that set of rules, so we can just make some assumptions.
Personally I believe the original idea was just to give some funny characteristics to the language the characters used. After all, it does somehow make sense in-context: if you was a pony living in a pony only community, why would use a word like "hands"? Things like "everypony" would then have originated from that.
This brings us to the second part of the problem. Do any proof of the existence of rules exist in the fictional word as portrayed in the show? While the question itself points out that there are multiple instance of "...pony" being used to reference a mixed group of creatures a recent episode seems to prove a rule is supposed to exist.
In the premiere of season 8 there is a specific line said by Applejack. She uses the term "everypony" to refer to a group of students in the newly founded "School of Friendship". She immediately correct herself since the group of students does indeed include other characters that are not ponies (dragons, yaks, changelings and so on).
Applejack: Everypony— I mean, everyone, go to your next class! I just can't believe it. Fights breakin' out when they're supposed to be learnin' friendship?
Without going into un-needed story spoiler, the situation does indeed degenerate from that point onward - mainly due some rude remarks a character makes about the school accepting non-ponies students.
Neighsay: Those are students? But you said you were opening this school to protect Equestria! To defend ponies from... dangerous creatures who don't have our best interests at heart!
Neighsay: Well, perhaps you should return to your kind.
It is somehow clear from the actual plot that those statement should be regarded as a metaphor for racism and other discrimination-related issue. Anyway, forgetting that for a second what is relevant to this answer is that the above statements causes a sort of "diplomatic incident" with the exponent of the other races.
And when Princes Celestia herself has to mediate with those delegates... she makes an interesting choice of words:
Princess Celestia: Please, everycreature! If you can just explain what happened...
It isn't clear if that is even an actual word. However it is clear that in this instance she decided to purposely avoid the word "everypony".
Taking this in consideration, I think it is safe to assume that "everypony" is just a word the ponies are used to use because they mostly live in a pony-only community with few instances of interaction with other races. The few instances of them using "...pony" wording to refer to mixed groups could somehow be explained as simple speaking habits.
I would therefore say that the ponies probably use ...pony forms just because it is more natural for them. That said, they probably consider it rude to call someone a "pony" when he/she clearly isn't, in a way not too different of calling someone a "guy" when she is obviously a "girl". That is why they probably prefer to use more "inclusive" term when speaking to someone they don't really know, to avoid offending anyone.
UPDATE: also in season 8, in the episode "Surf and/or Turf" we can hear some hippogryph use a variant of the "everypony" expression - something similar to "everygriffon". This also reinforce the idea that there is some sort of in-world rule to these grammatical constructs. Probably it is safe to consider such forms as simple word that characters uses when talking about a group of creatures of the same specie.