At page 9 in Sean Carroll's book Spacetime and Geometry he states:
The proper time between two events measures the time elapsed as seen by an observer moving on a straight path between the events.
He then goes on to explain the "twin paradox". He describes two observers in a coordinate system, one at fixed spatial coordinates and one that moves away with a certain velocity and then comes back to the origin. He marks three events: $A$ as the starting point when the moving observer starts to move away, $B$ as the turning point for the moving observer when he start to turn back to the origin, and $C$ as the point when the moving observer arrives again at the origin (but at different coordinate time $t$).
From the definition that he gave the time measured by the stationary clock is equal to the proper time between events $A$ and $C$ – this is in accord with the definition of proper time between $A$ and $C$ as he defines it above. Then he computes the proper time between $A$ and $C$ as measured by the moving observer. Since the observer is moving, its trajectory is clearly not a straight line between $A$ and $C$ and thus, according to the definition above, we cannot interpret the proper time of the second observer as the time it measures on its journey. But this is exactly what he does to explain that the moving observer has aged less than the static observer.
Is the interpretation he gives for the proper time between two points wrong? Should it be interpreted (as it is on Wikipedia) as:
The proper time along a timelike world line is defined as the time as measured by a clock following that line.
With this meaning for the proper time, the explanation he gives for the apparent "twin paradox" makes sense.
Best Answer
A more-complete quoting from Carroll's book will clarify your question on his use of terms:
So, here, he is referring to the "spacetime-interval between a pair of [timelike-related] events"... which could be called the "proper-time interval of this pair of events" (akin to the magnitude of a displacement vector).
In this usage, proper-time (or probably more correctly "proper-time-interval") is a function of a pair of events $\tau(A,C)$.
Then,
in the next paragraph, he uses the proper definition of "proper time" (as you see in Wikipedia) in the parenthetical part ...
***[I think proper time in this parenthetical section should have been bolded].
In this usage, proper-time (as Minkowski intended) is a function of a worldline segment between events $\tau(\gamma_{AC})$.
Then, Carroll describes the Clock Effect ( as featured in the Twin Paradox),
with a conclusion starting at the end of p.10
So, the proper use of "proper time" is used (but not fully defined) in parenthesis***,
which is unfortunately not-bolded like its use in defining the "interval".
Here is a related discussion where I contributed an answer:
Equivalence of two definitions of proper time in special relativity