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When I started teaching, I was asking the same question which every one asks:
What area  is cross section. Internet was not  yet invented. Combed through books
Found an excellent treatment in
R. D.Evans "The Atomic Nucleus" Tata McGraw Hill Publishing Co. (1955)
You always learn something extra from Masters..

As a bonus found definition and discussion cross section for waves.
Also when you go to quantum theory, since everything has dual nature.
You should ask which definition should be used Evas tells you that you can use any one.
Both are consistent when dual nature is taken into account.

How do you  measure total cross section?  A newbie, or some one who is a hardcore theoretical physicist,
far away from realty of experimental world, would think  measure differential cross section and "sum" up over all
solid angles. Well well, this is not the way total cross section is measured. Again Evans devotes a few pages to this.
In the process you get a better understanding of cross section.




It must be emphasized that the computation of cross section in classical Physics
requires only and understanding of definition of cross section and equation of the orbit.




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