Interpretation of proper time in Carroll's Spacetime and Geometry

Question

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.

Answer 1

A more-complete quoting from Carroll's book will clarify your question on his use of terms:

(pg. 9)
The fact that the interval is negative for a timelike line (on which a slower-than-light particle will actually move) is annoying, so we define the proper time $\tau$ to satisfy $$ (\Delta \tau)^2=-(\Delta s)^2= -\eta_{\mu\nu} \Delta x^\mu \Delta x^\nu.\qquad(1.17)$$ A crucial feature of the spacetime interval is that the proper time between two events measures the time elapsed as seen by an observer moving on a straight path between the events.

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 ...

(pg.9) A crucial fact is that, for more general trajectories, the proper time and coordinate time are different (although the proper time is always that measured by the clock carried by an observer along the trajectory).

***[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 $\tau(\gamma_{AC})$.

Minkowski said (on the page between Figs. 2 and 3 in "Space and Time"):
If we imagine at a world-point $P (x, y, z, t)$ the world-line of a substantial point running through that point, the magnitude corresponding to the time-like vector $dx, dy, dz, dt$ laid off along the line is therefore $$ d\tau = \frac{1}{c}\sqrt{ c^2 dt^2 - dx^2 - dy^2 - dz^2}.$$ The integral $\int d\tau = \tau$ of this amount, taken along the world-line from any fixed starting-point $P_0$ to the variable endpoint $P$, we call the proper time of the substantial point at $P$.

Then, Carroll describes the Clock Effect ( as featured in the Twin Paradox),
with a conclusion starting at the end of p.10

(pg. 10 last word) An
important distinction is that the nonstraight path has a shorter proper time. In space, the shortest distance between two points is a straight line; in spacetime, the longest proper time between two events is a straight trajectory.

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".

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Here is a related discussion where I contributed an answer:

Equivalence of two definitions of proper time in special relativity

Answer 2

I suppose he means that there are two periods of proper time for the travelling twin (ie the period between A and B and the period between B and C) which, if added, are less than the proper time of the stay at home twin between A and C.

Answer 3

The definition he give in the book is correct and the same as the one found on Wikipedia.

My mistake in understanding his explanation came in the fact that he said a "an observer moving on a straight path between the events". This straight does not refer to the fact that the path is a straight line on a spacetime diagram, but rather, to the fact that the observer is inertial, i.e. its motion is unaccelerated.

With this in mind we clearly see that the path of the second twin is actually a straight path. And on the next two pages, he goes on to define the proper time on paths that are not necessarily straight (by considering the infinitesimal spacetime interval). This means that the proper time is actually the time between two events as measured by an observer on his path between the two events (not necessarily a straight one this time).