Gallian states : "a plane symmetry of a figure F in a plane is a function from the plane to itself that carries F onto F and preserves distances; that is for any points p and q in the plane, the distance from the image of p to the image of q is the same as the distance from p to q."
Now practically speaking, by 'preserving distances' do we mean not causing any deformation to the figure? And does 'carrying F onto F' mean that the region which the figure F occupied before the motion has to be the same after the motion? Is this correct? And is there anything else that is required?
I said motion because I was trying to think of it practically.
I would say that "not causing deformation" is a lot more vague than "preserves distances." You may or may not feel that scaling the plane with the transformation $x\mapsto 2x$ causes any deformation in the sense of changing shapes, but it does affect the size of the image. The same could be said for a reflection... it's just not clear what "deformation" means.
By stating that it preserves distances, we know that it is made up of rotations, reflections, and translations.
It means that every point of $F$ is mapped to a point of $F$, and that every point of $F$ has some point of $F$ mapping to it. I think that's what you're saying, probably.