In several sources such as the Wikipedia articles on conic sections and the matrix representation of conic sections and p.104 of Pettofrezzo's Matrices and Transformations, it is stated that we can rotate the coordinate axes when handling conics. For example, an appropriate rotation of the coordinate axes can result in the $xy$ term being conveniently eliminated. Another example is the result that states that the discriminant of the conic is invariant under rotations of the coordinate axes.
My question is: why do we express these ideas in terms of changing the coordinate axes instead of rotating the conic around the origin while keeping the axes fixed? It seems to me that rotating one object would be preferable because changing the entire coordinate system would mean that we have to change the coordinates of every object with which we are working on the plane.
The two notions are equivalent: Rotating the axes with respect to the conic is equivalent to rotating the conic with respect to the axes in the opposite direction.
Also note that rotating "the entire coordinate system" is also just rotating one object; the union of two lines, the axes. And it just so happens that the union of two lines is a (degenerate) conic, so really you are just rotating a conic in either case.