One thing which always bothers me is the example you use and everyone else states. If you say that you move at a considerable fraction relative to earth, thus your clock moves slower, I (the one on earth) say that I am moving at a considerable fraction relative to you, thus my clock moves slower
This is the most common misconception ( assuming I understood what you mean)
The key point to grasp the theory of relativity is the concept of symmetry
Newton and his maid(his word) gave for granted the existance of a hierachy of observers
A ship is in motion relatively to the earth
The earth is in motion relatively to the sun
The sun...
Newton thought that this chain must have a beginning
An entity at absolute rest (God)
This intuitive but false assumption made classic mechanics a bloody mess, despite of its success, which Newton himself and the others after hime , were aware of
Einstein cleaned it up claiming that all observers are equivalent thus he wrote a set of equations which are valid for any observers
Einstein's logical reasoning entails a non intuitive but true conclusion : There is not an absolute time
Coming to your doubts
The twin on the earth claims that the time on board flows slower
However also the pilot claims that the time on earth flows slower than the time on board
Otherwise you would break the principle odf symmetry
