Definitions of "linearity" across branches of mathematics or levels of math education

Not exactly an solution to your question, but frankly it's too big to be fitting a comment. The distinction as to exactly what is being linear is very important. It is possible to approximate non-linear behaviours if we are allowed to make the linear space (as in linear algebra) large enough. Approximating solutions to smaller dimensional non-linear equations by large dimensional matrix equations or least-squares problems.

One of possibly simplest examples would be a binary decision. Something should "activate" once temperature goes above some level, and we have an in-signal (variable in-value) saying "increase", "decrease" by one "notch" or to "stay the same". We would like to activate for example a fan if things are getting hot enough. So our function value is 0 or 1. Such a "step" function (like the Heaviside step function) is highly non-linear, but we can still describe it with a (high dimensional) linear state space and realize it with, say a slightly modified cyclic permutation matrix $\bf P$ (it should not be allowed to "flip around" but rather be "saturated" at the max and min temp ends). Then counting up would be multiplication with ${\bf P}^1$, counting down ${\bf P}^{-1}$ and staying the same ${\bf P}^0$.

EDIT Oh bollocks, I forgot the punch-line. Our out function would then be scalar product with ${\bf v} = [0,\cdots, 0, 1, \cdots ,1]$ to select the appropriate function value. ${\bf v}^T{\bf s}$, where $\bf s$ contains the state and $\bf v$ the function values and the $\bf s$ is updated as ${\bf s} = {\bf P}^k{\bf s}$ for the in-value increase (k=1), decrease (k=-1) or stay the same (k=0).