The hospital was in the town I lived in. It is now residential condos. The hospital, not the town.
I was born at a hospital in Manhattan, but my parents lived in the South Bronx. When I was about 2 months old we moved to NJ where I spent my life through college. . I consider myself a Jersey guy, but always say I was born in Manhattan because I think it’s cooler somehow.
I am truly a native of Cleveland, so I give that city’s name if asked where I was born. However, I was never a resident of the Cuyahoga County seat. My parents lived in the suburb of Parma, then moved to another suburb (North Olmsted) when I was three years old. I lived there until I moved to Lafayette in 2001.
If asked where I’m from or where I grew up, I say “around Cleveland”, then explain further if (as often happens) my interlocutor has spent time in Northeast Ohio. When asked “East Side or West Side?”, I know I’m in for a bunch of reminiscing and comparing (the answer is “West Side”, as any Clevelanders reading this post already realized).
I was born in a hospital in White Plains, NY, but my parents were living in a town called Brewster. I was only a year old when we moved out of New York, so if you ask me where I’m from, I’ll say Michigan.
I sometimes claim I was born in Austin.
In actuality I wasn’t. I am by birth a Yankee. 
But my father’s a Texan and I’ve lived most of my life in Texas and most of my time in Texas in Austin. And we moved here before I was old enough to speak.
When it becomes an issue, though, the fact that I was born in a very big city helps somewhat.
Not the birth part, but the “And where have you lived/where did you go to school?” question works more for me here. I can name the cities and states I lived in, though some of the cities/towns are not so well known. I could name the town where I went to high school but few people other than Texas history/geography buffs and people who lived in the area would know where I meant. A few people have driven through it on their way somewhere else, though at one time it was apparently a very important shipping lane. Before Houston became a port, my town was on the map. It’s on the earliest maps I’ve seen of Texas, but it’s just a tired little town now.
So when people ask me where I went to high school, I don’t explain the storied history of the place. I say "Down by (city it’s close to) and they usually take that. 
If someone asks excitedly what school I went to, they used to live there, I’ll elaborate.
I usually say “Near Athens” when asked. I was born at the hospital in Athens, Ohio, but grew up outside of the town in a little truckstop-village about 45 minutes away. No one has usually heard of it, unless they are from the next town over or something, so I refer to it by county if not Athens, but that is still slightly an unheard name, even for other people in Ohio.
Brendon Small
Born in Philadelphia, lived there for about a year, then moved to Croydon which is outside of Philly. When I left Croydon I was about 12 or so moved to Florida. Met the hubby when I was 18 or so and now that he is stationed in NC I reside in NC.
Oo, Brewster! We used to play golf there when the parents lived in Danbury. It’s my favorite exit on 84 
. Or is it 684 by then?
When filling out a form, I give the town where the hospital was. That’s what it says on my birth certificate.
But when asked where I’m from, I give the actual town I came home from the hospital to, and where I lived the next 18 years. (Actually, I usually give the general geographic area, cuz no one’s heard of my home town.)
Yeah, lots of people answering a question that wasn’t asked. To me it’s pretty straightforward - your birthplace is the place you were born, even if that is a hospital in a town you’ve never otherwise been too.
It is straightforward - people ask where you are born or they ask where you grew up. They can easily be two different places. I happen to have been born in a town in southern PA and also grew up there until I went to college.
The vast majority of people who ask me where I’m born are Japanese, so I start off big and go small. I was born in the States, Utah, and Salt Lake City, which happens to be both where the hospital was and my home town.
If I had been born in a separate town than my home town, and if someone from the States who asked, I would probably tell the town where the hospital is, because I’m really literal about stuff like this.
For 15 years it had never even occurred to me that the hospital I was born in was across the river in another town. Wasn’t until I almost answered it wrong when filling out some government form that my mom corrected me.
I answer with the country, unless asked specifically, since most Americans don’t know Japan well.