Each observer has their own proper time measured by the clock in their rest frame. However, one man's proper time is not another man's proper time. Time dilation means that each observer will see the other observer's clock running slower (compared to their own proper time measuring clock). But everything is perfectly symmetric from either observer's point of view as long as the relative motion is uniform. You measure your clock ticking at the "normal rate" (your proper time) and you see the other person's clock ticking at a slower rate. Similarly the other person measures their clock ticking at the "normal rate" (their proper time) and they see your clock ticking at a slower rate.
This is all well and nice, but it gets interesting when the two compare their clocks after one of them does a round trip. This means that one of them necessarily had to accelerate and decelerate and was not in uniform motion (technically, was not on a geodesic). Now you have an opportunity to actually compare those two clocks and you'll always find that the person in uniform motion (in this case, the observer at rest on Earth) was the one whose clock has ticked the most, and hence aged the most.
The best way to understand this is to realize that the length of paths in spacetime is measured by the total proper time along that path (measured by that path-traveller's clock in their rest frame). One can show that the paths of uniform motion (geodesics) have that length maximized, so any path that deviates from a geodesic (because of accelerations), will necessarily measure a shorter total proper time after a round trip.
EDIT AFTER FIRST COMMENT:
Time dilation isn't the appropriate effect to consider in this particular problem -- length contraction is. In Nick's frame, a length contracted ship passes by at speed $v$. In Molly's frame, a point-object (heh) Nick passes by an uncontracted ship at speed $v$. Clearly, this should happen quicker in Nick's frame because of the length contraction. Thinking in terms of time dilation simply doesn't help here. Think from the point of view of each observer and it will be quickly obvious which effect to use.
The proper time is a fundamental invarient. By this I mean that all observers, regardless of their co-ordinate system, will measure the same value for the proper time. This applies to General Relativity as well as Special Relativity.
In SR the proper time is defined by:
$$ c^2d\tau^2 = c^2dt^2 - dx^2 - dy^2 - dz^2 $$
This immediately explains why the proper time is the time measured in the rest frame, because in the rest frame $dx$, $dy$ and $dz$ are all zero so the equation reduces to:
$$ c^2d\tau^2 = c^2dt^2 $$
or:
$$ d\tau = dt $$
The point is that I wouldn't start from the definition that the proper time is the time measured in the rest frame because this is a consequence of a much deeper principle. The equation I gave for $d\tau^2$ may seem a bit arbitrary, but it's a special case of the much deeper statement that:
$$ d\tau^2 = \eta_{\alpha\beta} dx^\alpha dx^\beta $$
where the object $\eta$ is the metric, which describes the structure of space time (strictly speaking we use the above equation to calculate the line element not the proper time, but this is a complication we can gloss over for now).
In the example you give, you can calculate the elapsed time on the plane without using the Lorentz factor. Indeed, you can derive the Lorentz transformations from the expression for the proper time.
Best Answer
Each object has its own proper time. If two objects meet each other twice, taking different journeys in between, then each one will experience its own amount of elapsed proper time between meetings, and they may be different. object $A$'s elapsed proper time describes how much object $A$ aged between meetings, and object $B$'s elapsed proper time describes how much object $B$ aged between meetings. The objects may age different amounts between meetings.
There is no absolute time, in the sense of a time that is the same for all objects. However, whenever two objects meet twice, they both agree about which meeting occurred first. In other words, they both agree about the sequence from past to future. But they can only make such direct comparisons when they meet. If we try to compare the timing of things that are far away from each other, things become more complicated, because then we need to account for the light (or other signal) that must travel from one location to the other so that we can actually make comparisons.
One of the keys to thinking about this is to think in terms of meetings between objects. Each object has its own proper time, which depends on how it behaved between meetings. When two objects meet, they can directly compare their (usually different) proper times.
Here's a related post, which is fresh in my mind because I just wrote it yesterday: https://physics.stackexchange.com/a/440209/206691.