A quarter-turn stair makes a single 90° change of direction at a landing. Size both flights, the landing, and the footprint, with live code checks.
A quarter-turn staircase turns 90 degrees partway up, using a flat landing at the corner rather than angled winder treads. It is the same geometry the building trades also call an L-shaped stair — the two terms are interchangeable. "Quarter-turn" describes the 90° rotation; "L-shaped" describes the plan outline.
Yes. Both describe a staircase that turns 90 degrees at a landing. The calculation is identical.
At least as deep as the stair is wide, and never less than 36 inches in the direction of travel under the IRC.
A quarter-turn uses a flat landing to turn; a winder uses pie-shaped treads.