Okay let’s put this to rest. I get that you do not feeeeel like Columbia is in Harlem, and yet it is still certainly not misleading to say that Columbia is in Harlem because they do indeed have a shared history, culture, identity and geographic region. When Columbia makes a decision about it’s campus, it must interface with Morningside Heights and Harlem. When students go to bars they either go downtown, stick close to campus, or go to Harlem. Yes, Columbia is in a wealthier part of the Harlem area. But in terms of describing where it is in the city, saying it’s in Harlem is an extremely effective way to describe it. They share architectural styles, schools, community centers, parks, subway stops, etc. The boundaries on a map are distinct and yet the boundaries in real life are quite porous. Is Yorkville technically part of the Upper East Side? No. And yet, if I were to go to yorkville, I would tell my friends I was going to the upper east side. Why? Because they share so much in terms of history, culture, and identity. And because I’d be taking the 6, god help me.
Columbia, morningside heights and harlem are so deeply intertwined in every single way. For those reading, some people clearly don’t recognize that. Yet, search “Morningside heights” on google maps and you will notice that it basically encompasses exclusively Columbia’s campus and land owned by the school, right next to good ole Harlem. So yes, as has already been conceded, Morningside heights is technically distinct form harlem as every university with a campus is distinct from the surrounding community… and yet it’s right up there in harlem.
Some might even call it the gem of west harlem 
http://www.nydailynews.com/archives/opinions/columbia-gem-west-harlem-article-1.583922
(i’m aware this is the ny daily news)
Some people might write articles about Columbia and Harlem being “neighbors.” I like the term neighbor because while it makes it clear you share a neighborhood, it emphasizes that you aren’t living in the same house. Just like Columbia and Harlem share a neighborhood and can still be in two technically disparate places on a map:
http://spectatorarchive.library.columbia.edu/cgi-bin/columbia?a=d&d=cs20031009-01.2.3
Part of the north side of morningside heights is actually included in District 5 of the NYC School districting zone and a school affiliated with Teachers College of Columbia University sits squarely inside of it:
http://insideschools.org/districts/manhattan/district-5?filter_borough=1&filter_district_neighborhood=district_5
You could easily post article after article that reaffirms the idea that while technically in morningside heights, Columbia is in the Greater Harlem area and the connections between the two are very real and undeniable.
But we’ll just have to agree to disagree on this…
And as an aside, Hanover/Vermont comparison isn’t great for a multitude of reasons that aren’t worth going into. And yet, Hanover NH and parts of vermont manage to share a school district as well (https://en.wikipedia.org/wiki/Dresden_School_District) … oh the horror