When I say “Planet of the Apes”, I’m quoting someone else, not speaking for myself.
Without any other context, when I use the term “ape”, I am including humans. If I don’t want to include humans, I will generally say “non-human apes”.
But, is the Pope a primate?
I’m glad I read the whole thread before popping in to mention this great Stephen Jay Gould title.
As long as I can claim in good faith to be an absurd-sounding animal operating machinery, I’m in a happier place. 
Your phrasing “apes, chimps, gorillas and us” seems to suggest that you think that “ape” is the name of a species. It’s not. “Chimps” are a species (or two species, if you include bonobos), “gorillas” are a species, and “humans” are a species, and all of those together plus a few more are kinds of apes. In turn, apes are a kind of monkey, and monkeys are a kind of primate, and primates are a kind of mammal, but there are mammals that aren’t primates, primates that aren’t monkeys, and monkeys that aren’t apes.
Only if you insist that the “monkey” element of the terms “Old World monkeys” (which are catarrhines, like apes) and “New World monkeys” (platyrrhines) has to define a clade. You could also just allow that “monkey” is, like “zebra” above, a description, not a classification.
Wikipedia says they are, cladistically speaking.
But they aren’t typically referred to as such because the words “bird” and “dinosaur” have such wildly different connotations. When the need arises to refer to birds and dinosaurs by the same name, a thing which presumably happens only in scholarly articles, one could use Dinosauria, or any of further successively narrower terms all of which include the Aves.
Just so with what we normally label as humans, gorillas, chimpanzees, and orangutans. When scientific accuracy is called for we can call them all Hominidae, a term which was once restricted to only H. sapiens and our ancestors since our line split off, which was then thought to have happened much longer ago than it is today, on the order of 25 or 30 million years.
If I understand your argument, then I don’t see the problem. The cladist calls your uncle a hominid, a taxonomic designation which happens to include three or four other species of large primate who share a large amount of genetic material with your uncle. A primatologist wouldn’t call your uncle an ape because of the confusion that would be sure to ensue.
But there’s a question of degree. We are far less Old World monkeys than we are great apes, because the latter are much closer to us genetically. That’s what taxonomy does: it not only describes various groups to explain what is related to what, but also to indicate varying degrees of closeness. We’re also the first ancestral lobe-finned fish that crawled onto the beach, but far less than we are catarhhines. (Obviously we’re not New World monkeys at all.)
Taxonomy really only requires shared characteristics, not degrees of closeness. That’s why Giant Pandas used to be classified along with Red (or “Lesser”) Pandas - because they shared characteristics, despite being not-that-closely related (the relatedness being merely assumed because of the shared characteristics). Really, taxonomy is just gives things names.
Systematics, on the other hand, strives to determine the relationships between organisms; cladistics, of course, being one of the major systematic methods these days. And, since cladistics is hierarchical, there is no “less” involved; we are every bit as much of a sarcopterygian as we are a catarrhine. We are, however more closely related to other catarrhines than we are to other sarcopterygians.
I don’t really have much to add, but I wanted to note that this thread is awesome.
Also, OP, it might be worth pointing out to your anti-evolution friend that we evolved from fish, but there are still fish.
And then give him one of those Darwin Fish magnets. The one with feet. I love those things!
So if I tell everyone that I’m a bacteria, is that in some sense true since we all evolved from bacteria, even though the phylogeny/clades/whatevs don’t track that way?
No. We (as in Eukaryotes) share a common ancestor with bacteria, but we did not evolve from them.
Edit: Or, in some phylogenies, Eukaryotes share a common ancestor with Archaea, and Eukaryotes + Archaea share a common ancestor with Bacteria.
We are not “fish” because we evolved from fish. We are “fish” because there are things that everyone calls fish (lobe-finned ones) that are more closely related to us (as tetrapods) than they are to other things everyone also calls fish (raw-finned fish, sharks, etc.).
I call my pet cockatiels dinosaurs. If you’ve ever seen a hormonal parrot beak-threatening you for coming too near the place he’s picked out for nesting, you’d instantly understand the dinosaur connection. There’s a positively Jurassic contempt for mammals somewhere deep in those eyes.
I have cockatiels too, and I am frequently reminded of velociraptors. They particularly dislike my glasses - probably because they protect the tasty eyeball meat.
This is true, but again, as with “monkey,” only if “fish” is interpreted as a classification, rather than a description.
To use the word “fish” with taxonomic rigor, and avoid including yourself, you basically have to specify to the class—
“ray-finned fish” (actinopterygians, the majority of living fishes),
“lobe-finned fish” (sarcopterygians, coelacanths and lungfish),
“cartilaginous fish” (chondrichthyes, sharks and rays), or
“jawless fish” (hyperoartians, lampreys; and myxines, hagfish).
Only if it interpreted as a monophyletic classification. Classifications needed be thus.
Wait, isn’t Sarcopterygii a class of superclass Tetrapoda?
I take it you meant “need not,” there. And I disagree; when I say “classification,” I mean rigorous, systematic classification.