Here's an example of US usage (and a true story).
In my childhood home, there was a landing on the stairway between the first and second floor. A landing is a small, flat, level platform. In order to fit the stairway into a small space, it was U-shaped. A flight of stairs led from the first floor to the landing, then a second flight of stairs led from the landing to the second floor. You walked up the first flight of stairs, took a couple of steps across the landing, and walked up the second flight of stairs.
Therefore, the "stairway" had "two flights of stairs" and a "landing."
I've never thought about "stairway" and "staircase" before. I think we use both in the US and I don't think there is any difference in meaning.