Proper Time Interval

Hello. I'm having trouble figuring out from which perspective to measure the proper time interval for Special Relativity. In the text book, the definition says it's "the time interval between two events measured by an observer who sees the events occur at the same point in space." But in the derivation of the formula for time dilation, the authors use an example with two observers, one on a moving train, one stationary on Earth. The observer on the train measures the proper time interval as being the time it takes for a beam from a flashlight to go up (directly vertical), bounce off a mirror on the ceiling directly above the observer and return to the flashlight. But as I see it, the observer is not viewing the two events (the light leaving the flashlight and the light returning to the flashlight) at the same point in space because they've moved down the track and are in a different position from when the light left the flashlight. If anyone can help me with this, I'd appreciate it. Thanks.