Your first link is a restaurant in a shopping mall. It has the interior facade of being a diner, and it serves...avocado bites, spicy chicken nachos, kimchi burgers, etc. Not really the same!
Every now and again someone will open a "American Diner" here in London, then have normal opening times and serve basically the same food every pub serves, only with more milkshakes.
Like, no. I want my American-style hash browns, over-easy eggs, and country-fried steak, not the same burger every pub on the street is doing.
And (refillable) filter coffee please, not just espresso drinks.
Diners are something else. In Germany we have "American diners" where you pay for each cup of coffee.
It's not the same.