OK. There have been many occasions where I have been asked just what my problem is and after much thought I think the simplest way to elucidate it is like this:
Everything about relativity and time dilation, proper time, coordinate time, world lines etc. makes perfect sense; however ...
Take a moving clock with a two second tick - (e.g. a light clock with the mirrors 1 light second apart)
It travels for 10 seconds at 0.6c, traveling 6 light seconds, from event 1 to event 2.
This inertial clock will tick 5 times in that journey. The light will be reflected from the distant mirror 5 times and will arrive back at the clock's base, for the fifth time at event B. So we know that, measured from the frame of the clock, 10 seconds elapses between event 1 and event 2; that the clock travels inertially for 10 seconds covering a distance of 6 light seconds at 0.6c.
However, as shewn in this diagram from Wikipedia, where in our scenario
Δt = 2 seconds;
Δt' = 2.5 seconds
1/2 v Δt' = .75 light seconds which gives us 7.5 light seconds between events 1 & 2 in 12.5 seconds.
Measured from a stationary frame the traveling clock takes longer and travels further.
View attachment 208094
Now the problem that I have is that it is the same journey between events 1 & 2.
10 seconds measured for the inertial clock by the observer traveling with the clock is proper time - time measured between two events on the clock traveling inertially between those events.
12.5 seconds measured by the stationary observer measuring the time passing for the moving clock in the stationary frame.
In the Lecture in which he introduced his Spacetime theory, when discussing length contraction, Minkowski said
The distance the clock travels is the same = 6 light seconds.
The proper time, measured by/recorded on, the clock between the events is 10 seconds. (specified in the description)
The coordinate time for the moving clock is longer = 12.5 seconds measured by the observer at rest in the stationary frame. This is the total of the time measured internally by the clock, the proper elapsed time, and the time taken to travel the distance from event 1 to event 2. (Added as vectors - simple pythagorean triangles)