No; the only case in which a "change in direction" is considered an eigenvector is when the change in direction is reflection about $0$; that is, when we send $x$ to $-x$.
The definitions are not equivalent: the rotation of $\mathbb{R}^2$ by $90^{\circ}$ counterclockwise around the origin maps every vector to another vector of the same magnitude, but does not have any eigenvectors.
A matrix $A$ with the property that $\lVert Ax\rVert = \lVert x\rVert$ for all $x$ does have a special name: they are called unitary transformations when the vector space is complex, and orthogonal transformation when the vector space is real.
The reason you don't want to call what you have an "eigenvector" is that eigenvectors are useful ways of decomposing transformation into easily understood pieces. The kind of result you are describing does not help in doing that.