This is a common confusion students have with coordinate systems when first learning SR. They hear us say things like "the time according to observer Bob" or "in the reference frame of Alice" or "boost to the rest frame of Charlie". To be clear, these are short hand. Something isn't literally/physically "in" one frame and not "in" another frame. When we refer to the coordinate system "of" an observer, or something "in" someone's frame, it is usually short hand for discussing values according to the instantaneous co-moving inertial coordinate system.
So do not take such short hand literally.
A coordinate system is just a systematic labeling of spacetime points. We can choose any coordinate system we want. As an example that will hopefully get past this intuition barrier, choose a coordinate system in which you are at rest. Now change the coordinate system by adding a constant (shifting the origin). You don't need to physically move to use this new coordinate system.
In short, nature doesn't care how we label spacetime points. So we should be able to explain the physics independent of choice of coordinate systems. This is indeed possible, so you can really use whatever coordinate system you choose. A particular choice may be simplier to work with based on your motion, but no choice is more mathematically correct than any other.
As should be apparent from the other question you asked, the Motion Mountain book is not reliable. Just searching in Google brings up the phrase crackpot in link titles on the first page of results. Please stick with reputable sources.