Was Hogwarts named 'Hogwarts' because of a connection to warthogs?
Or was it named for hogs and warts separately, or something else entirely?
Out of universe: There's no canon proof it has any connection to warthogs; the Word of God answer is either "Not knowingly" or "no" depending on an interview.
In-Universe, a later ret-con by JKR (which is nevertheless, canon) actually suggested the reason, and your "named for hogs and warts separately" was ironically right on the nose:
The location and name of Hogwarts were both chosen by Rowena Ravenclaw, who dreamed that a warty hog was leading her to the cliff by the lake. (source: W.O.M.B.A.T. Grade 3 Quiz on JKR's website posted in June 13, 2007, copied on HP Lexicon here)
Now, a "warty" hog is somewhat related to warthogs biologically, but the latter species only live in Africa and presumably would not have been leading Ravenclaw to a location in Scotland, or even known to a British witch of the time.
Details for out of universe naming:
The only 2 mentions of Hogwarts name origin by JKR were in an interviews:
Q: How do you come up with names?
A: Some I make up. Some mean something. Dumbledore is olde English for bumblebee. I thought I made up Hogwarts, but recently a friend said, 'Remember we saw lilies in Kew gardens (a garden in London.)' Apparently there are lilies there called Hogwarts. I'd forgotten!
(Abel, Katy. "Harry Potter Author Works Her Magic," Family Education website, undated (October?) 1999)
Note that the actual plant name is "hogwort", spelled differently.
2004 interview that has a slightly different note:
Bradley y4 Griffyndor: Why did you call the school Hogwarts?
JK Rowling replies -> I tried all sorts of different versions of the name and then this word floated into my mind and I knew it was the right one.
(JK Rowling's World Book Day Chat, March 4, 2004)
Note that she didn't elaborate on whether the reason the word came to her was warthogs or not....
Having said that, interestingly (and perhaps showing that it may have subconsciously the connection did influence her) she does think of warthogs in connection to Harry Potter at least once. From the same interview in 2004, amusingly:
kelly_holland: When you turn into an Animagus, can you choose what animal you become? Or does this get "assigned" to you?
JK Rowling replies -> No, you can't choose. You become the animal that suits you best. Imagine the humiliation when you finally transform after years of study and find that you most closely resemble a warthog.
Another interesting circumstantial evidence is that other magical school names are also meaningful phrases composed of two parts: Beauxbatons and Durmstrang
There appears to be no direct canon confirmation of a warthog link at all: I have looked through every single book, other interviews, and other sources, and the ONLY mention of Warthogs in the whole of canon outside that last interview quote is the (probably NOT JKR-initiated) fact that in the Chamber of Secrets video game, Rubeus Hagrid said that he would like to own a Warthog. That's NOT in the book, just the game.
JKR claimed that the word "floated into her mind" but failed to say where from. There's ample evidence that she was inspired by the film This is Spinal Tap regarding at least one other aspect of the books (the curse against the DADA role) and which also contains this line
Ian: The specific reason why he was knighted was for the founding of Hoggwood, which is a summer-camp for pale, young boys.
This sounds like a perfect description of Draco Malfoy (who is always referred to as a pale boy) and the word 'Hoggwood[s]' is not a million miles from 'Hogwarts', linguistically speaking.