Confusing definition of proper time – which is correct? I have googled for „definition of proper time“
This source https://www.collinsdictionary.com/dictionary/english/proper-time gives the following definition:


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*proper time ... measured by a clock that has the same motion as the observer.  Any clock in motion relative to the observer ...  will not, according to the theory of relativity, measure proper time.


However,  according to this answer 
Confusing time dilation - proper time is higher?
„In other words, it is the time registered by a clock that is carried from one event to the other“ exactly the moving clock measures proper time interval and this time interval is the shortest due to time dilation. 
Is definition in Collins Dictionary wrong? Please help resolve this contradiction.
 A: The definitions are both trying to say the same thing, but they are not quite managing to avoid all scope for misunderstanding. For a non-technical appreciation of the meaning of proper time you should start with the principle that proper time is the time experienced at any point in one's own reference frame. As you sit at your desk marvelling at the clarity of my answer you are experiencing proper time in the reference frame in which you are stationary. Any clocks that are stationary relative to you will record time at the same rate you experience it.
Anybody moving relative to you will experience their own proper time, which will be faithfully recorded by any clocks moving with them (ie clocks that are stationary in that person's reference frame).
The Collins definition was insufficiently precise. It should have said that any clock moving with respect to an observer will not measure proper time in the observer's frame of reference. 
If a clock is moved between two other clocks that are stationary relative to each other, the time it records is a proper time for that clock's frame of reference, and it will be shorter than the time that appears to have elapsed according to the stationary clocks. 
A: you are right. A formal definition in textbooks (already in A.Einstein) is simple but technically and practically confusing; formal- that integration along the path (on a diagram) as contrary to a coordinate time in at rest observer (with a clock, rather two synchronized clocks at the distance x=vt). 1) Lorentz equations are symmetric in nature-when you switch coordinate systems: changing in the description from (x,t) to (x',t') you will get a reverse number for both times.2) The translation from one CS(coordinate system) into another is only a kinematical and not dynamical (as it is claimed in Twin Paradox Absurdity)- pointed rightly by M.Sachs.3)the confusion is already in A.Einstein, who does not at all speaks about any real clock but only about an ideal light clock that is just a model and nothing real! In a practical experiment with atomic clocks, the difference between "proper" and "coordinate" time arises from Relativistic Doppler Effect - such clock mechanism acts as a wave-field(counting the period or frequency wave of source emission. The same with "moving radioactive substance" called muons or others. The laboratory measures something "moving" in relation to the laboratory at rest system: this sth moving is not at all an object but rather a wave; consequently, one measures this change in the frequency of such substance. I know for sure: physicists, in general, do not know the right meaning of time parameter in Einstein's theory except jus a formal maninpulation of it in equations. The question arises: which is the "proper unit time"? If you use the same atomic clocks (as is assumed in both systems, at rest and moving one); then, a "unit time" must be applied to at test coordinate system and not to "moving" clock attached to the body" because such unit time would be changed according to Doppler formula; in any other clock, this talking is useless because the mechanism of such clock and its unit time (or period) does not depend on the velocity- "proper time is just an imaginary time ('it looks like less') than a normal, coordinate time recorded by clock/2-clocks in at rest coordinate system.
