I believe it is important to denote correctly, who is "at rest" and who is "moving". Quite often books and articles suppose, that if the events are "co-located", this observer is necessarily is "at rest".
There is a lot of confusion, who moves and who at rest, because according to this theory someone can be either "at rest" or "moving". An observer is at rest, but he is moving, for someone he is moving, but someone is moving according to him. This makes dizzy.
Let's someone traveled from point A to point B. There were two events: departure from A and arrival to B. These events were not co-located in the Earth frame, but co-located in the travelling observer's frame, because he had remained in the same car of the train.
An observer on the Earth (who is "at rest") measures interval of travel time with two spatially separated and synchronized clocks, an observer in the train measures travel time interval with a single clock. Hence, if the events for someone were co-located, this observer was the "moving one". His clock measured shorter interval of time, than two spatially separates clocks.
Wikipedia article about Motion claims: In physics, motion is the change in position of an object with respect to its surroundings in a given interval of time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, and speed. Motion of a body is observed by attaching a frame of reference to an observer and measuring the change in position of the body relative to that frame.
To move relatively to a single clock is not possible, because it "has no surroundings". Two spatially separated clocks create that very surroundings, and single clock moves relatively to them.
Single clock measures shorter time interval, than two spatially separated in Einstein – synchronized clocks. Two spatially separated clocks record greater time interval.
That means, that since single moving clock runs slower, time in the whole reference frame, in which this clock is moving, runs faster.
Time in the reference frame of “stationary” observer is the same in any point of his frame, it does not depend on coordinate, that goes out straight from Lorentz transformation for time.
Time Dilation Special Relativity