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.