Not my experience. It’s almost a cliche that Starbucks is where people plop down with a laptop for hours at a time, just to get out of the office if nothing else. I’d agree that it doesn’t have quite the same cafe culture vibe as independent cafes do, though I think that’s just resistance to it being a chain rather than being something about the way its operated. Locations play a role, too. A Starbucks with a bunch of foot traffic will be different than one in some depressing strip mall.
The USA, or at least New York City has a pretty big coffee shop culture. There are still a ton of coffee shops but since COVID a lot of them got rid of a lot of seating so they are less of a place for hanging out, organizing casual meetings and whatnot.
Which is a shame IMHO and probably one of the reasons Starbucks is having so many financial problems. Some Mckinsey study probably showed that people waste so many minutes sitting around the coffee shop when they could be using that time to buy more coffee. So they got rid of the seats but instead of buying more coffee people just said “fuck it” and didn’t go there at all.
I don’t know that there is a hard line between the two, but I believe cafes have wait service while coffee shops don’t.
Cafes also have more of an environment of people just sitting there at a table casually sipping their coffee and reading while coffee shops tend to be more “get my coffee and go”.
But again, the lines tend to blur. Especially when you start to throw diners and sandwich shops into the mix.
Mea culpa!
I’ve seen and utilized some of the drive through coffee shops. Aroma Joe’s near Pittsburgh is very tasty.
We had one of those just open in our slice of suburbia here. When we go by on Saturday mornings there are more than 100 cars lined up for AFAICT either two or four windows. I don’t care if they’re giving it away, I am not going to sit in my car behind 99 other cars just for a drive-through cup of coffee.
It may be worth noting that, especially compared to some European countries, the U.S. “coffee culture,” as it stands today, is a pretty recent phenomenon.
Growing up a Gen Xer in the 1970s and 1980s, coffee was something that our parents and grandparents drank. It was bitter, and nasty, and we couldn’t understand why anyone would choose to drink it. I knew virtually no one in my age cohort who drank coffee even occasionally, much less regularly.
Well, the reason was that the major U.S. coffee makers had increasingly switched from arabica beans, to the cheaper, but more bitter, robusta beans over several decades; older, lifelong coffee drinkers got used to it, but first-time drinkers were turned off by it.
Then, in the '80s and '90s, companies like Starbucks, Peet’s, etc., which started using arabica beans again, and sold coffee as a premium product, began to expand nationally, and demonstrated to younger consumers that coffee did not need to taste terrible.
In what I would call a “coffee shop,” someone will at least come over to your table/booth periodically and ask if you’d like a refill*. At a diner, you’re more likely to sit at a counter alongside a bunch of people you’ve never seen before. But, as you say, the lines are sometimes blurred.
*Does this give away my age? 
I haven’t lived in the US for more than 30 years.
In Canada, Tim Hortons is everywhere and there are many Starbucks. I find both make mediocre coffee, far worse than in any small Mexican town. But Starbucks has an attractive refill policy on basic drinks.
Quebec has Van Houtte, which is light years ahead in terms of the quality of pastries and coffee. Second Cup and Timothy’s were decent Canadian chains now either closed or rarely seen. Several decent independent places still exist, especially in bigger cities. Our West, there are chains like Blenz and Coffee Culture which are okay. I find Dunkin Donuts very mediocre. Kristy Kreme has great donuts - wish there were more in Canada.
For me, “cafe” and “coffee shop” are mostly interchangeable. I worked at a place called “Cafe Express” in the Chicago suburbs in the 90s. All of us referred to that place as either a cafe, a coffee shop, or a coffee house – it didn’t matter. It was a sit down place with counter service that served coffee and had some sweet pastries. It was a place where people would sit down, have a conversation, read a book, or otherwise while the time away. On Sundays, there may be live music. It served a similar sort of function as, say, a pub does in Britain: it was kind of an extended living room, except this one served coffee not pints. If there’s a difference between the terms “cafe” and “coffeehouse,” I struggle to find it. At least in this region, they are the same.
At a coffeehouse you can smoke Mary Jane, recite your own poetry, and listen to cool jazz, Daddy-O!
Where are my bongoes? 
Oops. I actually meant “coffee shop” at the end there, as I use that more commonly than “coffeehouse.” But, yeah, when in Amsterdam, “coffeehouse” does have a more specific meaning.
Where I am (SF Bay Area) Starbucks still dominates. There are only about a dozen Dunkin’s in the whole Bay Area. It’s true that over the last few years many Starbucks locations have closed, but there are still way, way more than any other chain. The closest is probably Peet’s, which has 200 locations in the US.
Oh yes I visited a few of those!
Actually, I was thinking more about California or New York. But Amsterdam is cool too.
You can’t smoke at coffee shops in NYC. Anyplace cigarettes are banned, so is marijuana.
Starbucks is the dominant coffee shop here in NYC. Looking on the map there are more than 20 locations less than 1 mile from me!
You can if you borrow @terentii’s time machine
Coffeehouses, not coffee shops!
I was born too late to be a Beatnik, so I have to make do with what I’ve got. 
These aren’t the top restaurant chains in the U.S. These are the top coffee shops/whatever. Here’s a list of the top restaurant chains in the U.S.:
I said exactly that; you clipped my post.
I used that same article which you posted, went through the list of restaurants myself, and pulled out of it the top coffee chains – and even gave their overall ranks among all restaurant chains, directly out of that article.
For example, I said that Dutch Bros is the #3 coffee chain in the U.S., and the #48 overall restaurant chain in the U.S., based on those rankings from that article.
My full post read (and I’m bolding where I said I filtered the list myself to only show coffee chains):