Time displayed on an observer's wrist watch...
On account of the effects of
time flows at varying rates for different observers. This requires that we be careful in specifying whose time we are talking about. The time displayed on the t-axis of a spacetime diagram is the reference frame time. Each observer in that reference frame experiences her "own time", "eigenzeit" in Einstein's native language. In English that is usually called "proper time". An observer's proper time is that displayed on a clock that accompanies observer throughout her relativistic adventures.
Of course the proper time will depend on the time set on the clock initially so in preparing a spacetime diagram it is customary to re-zero all clocks at the time origin of the diagram. From that time on, the duration of each observer's proper time unit as seen from the reference frame will depend on that observer's relativistic status.