A proper exposition of the effects implied by special relativity concentrates on the following: when your motion does not involve any acceleration then the amount of time that elapses for you is a maximum.
So: we have Bob not accelerating. Let's say he is moving in the sense that he is moving forward in time. Alice is on a trajectory that involves an acceleration phase away from Bob, and a U-turn in any way, and then rejoining Bob. At the instant of rejoining they are once again right next to each other, so they can make a direct comparison of how much proper time has elapsed for each of them.
On the other hand, there is ambiguity when they compare their clocks while Alice is far away from Bob, because then you have to take transmission delay into account. Likewise, comparing clocks while Alice is on the move is ambiguous, because then you have to take classical Doppler effect into account.
The effect that is specific to special relativity is laid bare when you remove all other effects. You set up a scenario where at the moments of comparison there is no transmission delay, and no Doppler effect.
(So anytime you encounter an exposition of special relativity where a scenario is offered that includes transmission delay and/or classical Doppler effect you can just stop reading that exposition; it won't teach you.)
So you focus on the scenario of Bob not accelerating, just moving forward in time, and Alice accelerating away, making a U-turn, and rejoining.
For Alice, by traveling more spatial distance than Bob, less proper time elapses. That is: there is something you can actively do so that for you less proper time elapses.
On the other hand, there is nothing that Bob can actively do so that for him even more proper time elapses; he is already moving forward in time along the spatially shortest path, so he is already maxing out the amount of proper time elapsing for him.
So it's not a symmetrical situation; Bob is already maxing out, Alice can make the amount of proper time that elapses for her smaller by increasing the spatial distance she travels (in the same amount of Bob's proper time.)