1/ I consider that the twin paradox is formulated as follows: "Two people in the universe cross each other, synchronize their clocks at this point in time, continue to travel in the universe and cross paths again. When they meet again, which one has aged faster?"
In that case, is it true to say that there is nothing in the formulation allowing us to answer the question, even if one of the people is called "the traveler"? Only a physics experiment could tell us which one is the "true traveler", that is to say the one who is not in an inertial frame? Furthermore, is it true that both persons could be in non-inertial frames, making the use of SR to solve the paradox very debatable?
2/ If we now consider the classical formulation, with one twin on the Earth, and the other in a rocket. Is it possible to imagine that the traveler is following the path of an object "free falling"?
This point needs clarification. If I assume that the Sun is in free fall in the universe, the Sun is the inertial frame needed to be able to apply the laws of SR to solve the paradox. I can (in thought) consider a little object linked to the Sun through a rigid link, and positioned exactly somewhere on the orbit of the Earth. This object should be, as much as I understand, in an inertial frame.
If we are now considering that the traveling twin is in a rocket (or a satellite) moving precisely like this little object (same position at any point in time), even if the rocket is not itself an inertial frame, the twin in it should (in my opinion) see the same things than those "observed" by the little object (or an observer sitting on the little object).
In that case, is it true that the traveler would age faster than the twin on Earth, as the twin on Earth would be the one "truly moving"?
If that is the case, is it correct to state that the result of the experiment when we consider the Earth and a rocket (both non inertial) depends (theoretically) on the motion of the rocket?
Edited to clarify my question after a discussion with the other contributors