A friend of mine who was born on an American military base in Japan says “American military base in Japan” when asked where he was born.
Where were you born? Worcester
Where are you from? Massachusetts, near Worcester
I was born in Worcester Memorial, but I’ve never lived in the city.
Where were you born? Scotland (Enough detail from this side of the pond.)
Where did you grow up? Sarnia. (Followed-up by South Western Ontario, if I get a strange look.)
If asked where I was born, I’ll say the city. If asked where I’m from, I’ll say Saskatchewan. Most people let it go at that; other Saskatchewanians might want to know where, in which case we’ll get into the longer discussion of which cousins and friends we have in common. 
I just say Texas. If they need clarification, I’ll kindly admit that it is part off the United States.
Usually, if someone asks where I was born, I tell them San Antonio.
If they ask where I’m from, I usually say Houston
And if they ask where I live, I say Dallas(even though I live in a suburb).
(I was born in San Antonio during my dad’s second-to-last year in the USAF, and after his enlistment was up(1973), we moved to Houston, where we lived until 1999, at which time I moved to Dallas.)
I know folks from TP, PH and Palos Park and even Naperville and they all say they are from Chicago!
For me it depends on where I am when someone is asking. If I am not in my home state then I tell the person I was born in New Jersey. Otherwise, I will actually answer that I was born in Newark. Interestingly, virtually no one in the town I grew up in was actually born there. Most people from my town were born at St. Peters in New Brunswick.
Same situation for me - I grew up in Alabama, but I was born across the river at a hospital in Columbus, Georgia. I usually give Columbus as the place I was born.
I just limit it to the country. I’ll only narrow it down to the city if it comes up later in conversation.
The question reminded me of the episode of King of the Hill where Hank Hill was horrified to find out that he was born in New York City while his parents were on vacation there. (For those not familiar with the show, the character strongly identifies as a Texan.)
Personally, I think it’s OK to identify your birthplace as the town where your family lived even if you were born in the hospital in a neighboring city.
Where were you born? “Kingston, NY.”
Where are you from? “Originally, New York State, Hudson valley area.” I say orginally because I don’t go back there and don’t identify with being from there except in limited ways (IBM brat).
Just the city.
Memphis.
If asked where I was born I say, “Chicago”.
If the person is a Chicagoan, I might say “Michael Reese Hospital”, which is in Chicago and they’d likely know that.
When asked where I’m from I nearly always answer “Chicago” even though I haven’t lived there in more than a decade, and have moved several times in that period.
The hospital was in a different town, so I would list that as my birthplace (it’s where my birth certificate resides). It was a bit galling in high school, since the other town was our big sports rivals.
I was born in Queens and grew up there, but I usually say New York which works for most of the country. My kids were both born at the Medical Center at Princeton. We actually lived in Princeton when the older one was born, but had moved to a small town for the second, but we usually say Princeton for her because it’s cooler to be born in Princeton.
I almost always say “Detroit,” as I spent the first 18 years of my life in the city proper (and I do always have to emphasize I was from the city proper. In Michigan, if a white person says they’re from Detroit, it’s assumed they actually mean one of the suburbs).
The only time I give the precise location of my birth (Grosse Pointe Farms, where the hospital was) is when it’s for governmental reasons. I figure they want me to be exact. That said, when I’ve sometimes blanked on the city name (there are five Grosse Pointes), I’ve said Detroit and there were no repercussions.
I smugly say that I was born in the same hospital - St Mary Abbot’s, in Kensington - where, not 11 short years later, Jimi Hendrix’s body was taken after his fatal overdose.
I say the name of the town of the hospital, (Bristol) since it’s old (for the states, the eurodopers would just laugh) and moderately interesting, whereas the suburb I grew up in, (Fairless Hills) is just boring.
Air Force brat checking in. If asked where I was born, I name the town nearest to the base we were living on and where the hospital was (my official records list this as well).
If asked where I’m from (or grew up) I say: “Nowhere, I’m a military brat.” (and its amazing how many people don’t realize “military brat” is an accepted term and I’m not saying I was a miserable child.)