It is not a stupid question. The definition above is exactly the definition of a linear function, i.e. a function f that satisfies f( cx + y ) = c f ( x ) + f ( y ) ( note that you can pull the scalar out, so that functions of the form cx are linear )
the map f ( x ) = ax + b is taught to us in grade school as a linear function on the basis that it draws a line. Actually though, it is called an "affine function" ( it acts essentially like a linear map though, the map will satisfy all the properties if you simply translate everything by b ).
In the context of differential equations, both cases are known as "linear equations"