$$a\cdot x_1+b\cdot x_2+c\cdot x_3+...+qx_n=\text{constant}$$ is called a linear equation because it represents the equation of a line in an $n$ dimensional space. So "linear" comes from the word "line". Basically there should not be any higher power of $x$ failing which the graph of the function will not be a straight line.
similarly
$$a(x)y+b(x)y'+c(x)y''+d(x)y'''+...+q(x)=0$$ is also called linear differential equation because all the derivatives have power=1 which is similar to the above definition of a linear equation.
A function f is called linear if: $$f(x+y)=f(x)+f(y)$$ and $$f(c\cdot x)=c\cdot f(x).$$ Here c is a constant. In this definition of linearity of function $f$ what does the word linear means? How does it relate to a straight line?
Finally what does the term linear means in case of linear vector spaces? Where is the reference to a straight line?
So, whether linear is just a word used in different contexts? Does it have different meaning in different situation? Or linearity refers to some relation to a straight line?
As you pointed out, linearity is the property of maps $f$ that satisfy $$ f(a x + y) = a f(x) + f(y)$$ for scalars $a$.
A vector space has a linear structure, in that if $x, y \in V$ then $ ax + y \in V,$ and the structure-preserving maps between vector spaces are the linear maps.
A linear map $f$ between (finite-dimensional) vector spaces can always be represented by a matrix $A$, i.e., $$ f(x) = Ax.$$
Also note that strictly speaking, the equation of a line $ f(x) = ax + b$ is not linear unless $b=0$, because a linear map must preserve the origin, i.e. map zero to zero. Only lines passing through the origin qualify as linear. General lines are examples of affine maps.