Does the scheduled departure time refer to leaving the gate, leaving the ground, or something else?
Similarly, does the scheduled arrival time refer to the landing, arrival at the gate, or something else?
Obviously, variations will be common but I presume that the times refer to some event.
The Departure and Arrival times are estimates for when a plane expects to start maneuvering away from the departure gate and stop at the arrival gate respectively.
Prior to a flight, a Flight Plan must be submitted. These plans contain a huge array of fields regarding the flight. The International Civil Aviation Organization publishes and maintains documentation (Doc 4444) regarding the flight plan, its content and format. There are fields reserved for Departure and Arrival times, specified as estimates.
Air traffic is very complex and dynamic and so a flight plan describes expectations but actual times and other data (speed, altitude, etc) can differ when the actual flight procedes. Many decisions are made based on these plans and so they strive to be as accurate as possible.
Does the scheduled departure time refer to leaving the gate, leaving the ground, or something else?
Typically it's when the cabin door closes or when the parking brakes are disengaged.
Similarly, does the scheduled arrival time refer to the landing, arrival at the gate, or something else?
Typically it's when the parking brakes are set.
Parking brakes are a popular choice since they can be tracked automatically and it has only limited amount of wiggle room for interpretation. "Gate arrival" and "Gate departure" can mean different things to different people. This is important since the exact definition can have legal and regulatory consequences (pilot hours, delay compensation, final decision authority, etc.)
Airlines determine these times simply by running statistics. They track the time on every route and look at the distribution of actual flight times (as defined above). They then choose a time so that maybe 80% will be on time and 90% within 10-15 minutes of on-time (depending on the shape of the distribution)
It's NOT the average or median flight time: otherwise half of the flights would be late. In fact its quite "normal" for a flight to be early. This also plays a factor in airlines setting minimum connection times which can sometimes feel ridiculously low.
What do scheduled flight times refer to?
Scheduled departure time from the gate and scheduled arrival time at the gate.
E.g., see Horizon Flight 2375, scheduled departure time is 4:15 pm: https://www.alaskaair.com/status/2375/2024-04-29a
and looking at the details, that matches the scheduled departure time from the gate on https://www.flightaware.com/live/flight/QXE2375/history/20241009/2325Z/KGEG/KSFO:
In the very specific context of EU's Air Passenger Rights during flight delays, the question was conclusively answered by the court ruling in Case C-452/13 (Germanwings GmbH v Ronny Henning). They list four possible definitions of "arrival time" that exist in various IATA and national air traffic regulations:
(a) the time that the aircraft lands on the runway ("touchdown");
(b) the time that the aircraft reaches its parking position and the parking brakes are engaged or the chocks have been applied ("in-block time");
(c) the time that the aircraft door is opened;
(d) a time defined by the parties in the context of party autonomy
And then rule that definition (c) applies when deciding whether the delay was large enough to trigger EU261 compensation:
... Articles 2, 5 and 7 of Regulation No 261/2004 must be interpreted as meaning that the concept of 'arrival time', which is used to determine the length of the delay to which passengers on a flight have been subject, refers to the time at which at least one of the doors of the aircraft is opened, the assumption being that, at that moment, the passengers are permitted to leave the aircraft.
