This is super frustrating. I literally cannot grasp the concept of what proper time is and I have no idea why. There's something I'm missing here.
If someone is standing on earth and there's a clock next to him in the same reference frame, and then there's someone in a spaceship traveling at $0.5c$ away from the earth with another clock in the frame of the ship, then from the view of the earth the clock on the ship will run slow. I get that.
But who measures proper time? I don't get it at all. The definition given everywhere makes no sense to me. My book says that the proper time interval is "the time interval between two events measured by an observer who sees the events occur at the same point in space." At what same point in space? For whom? What does this even mean at all? Can both the observer on earth and the observer in the space ship both have a proper time? By the time dilation formula it's obvious that it refers to the minimum time that's possible I guess. Would one the proper time in the ship be the improper time on earth? Would the proper time on earth be the improper time on the ship?
I'm literally hitting my desk just not being able to understand this and not getting a clear answer from anywhere.