intutive difference between linear map/transformation vs linear function

my linear algebr textbook defines a linear transformation/map as one that satisfies:

i. T(u+v)=T(u) +T(v). ii. T(cu) = cT(u)

However, what is traditionally called a linear function, in non-abstract algebra (or highschool algebra, or whatever it is formally called), namely: f(x) = a + bx is not a linear mapping according to the linear algebra definition, unless a = 0.

Is there an intuitive reason why the first definition is called a linear map, and why you would not call y=1+x a linear map, despite the fact that it defines a straight line on a plane? I could simply take the definitions as given, but it always helps me to have an intuitive understanding of such terminology.

the reason is that a linear function does not preserve the origin. but a linear map with the properties you listed does!

example of linear function: $$f(x)=a x +b$$ $$ \begin{align} f(u+v)=a(u+v)=au+b+av=f(u)+f(v)-b \neq f(u) + f(v) \end{align} $$

example of linear map: $$g(x)=A x$$ $$ \begin{align} g(u+v)=A(u+v)=Au+Av=g(u)+g(v) \end{align} $$

With a linear function you cannot transform a vector space into another vector space, thing that you can do with a linear map.

So now comes the intuitive way of seeing it: A linear map takes vectors and rotates and scales them and project them onto a subspace (not necessarily). A linear function does the same plus in the end it translates the origin, applying a translation distrupts many beatiful and USEFUL properties.

Remark: In general $x$ is column vector with N elements and $a,b,A$ matrices with K rows and N columns. But this example works in the one-dimensional case too ($K=N=1$)