The following is a modern, fairly general form of Bézout's theorem. (There are forms that are more general and/or more precise; bear with me.)

Define the degree of a reducible variety to be the sum of the degrees of its irreducible components. Then, for any two varieties $V_1$, $V_2$ in affine space $A^n$ (or projective space $P^n$), $$\deg(V_1\cap V_2) \leq \deg(V_1) \deg(V_2).$$

I've seen this result credited to Fulton-Macpherson (e.g., in Shokurov-Danilov). This feels too late; I imagine they are really responsible for a more general result. At the same time, I haven't seen this result clearly stated earlier in the literature, not that that means anything - I am no algebraic geometer. (As a friend just reminded me, Thm. I.7.7 in Hartshorne gives the special case of $V_2$ a hypersurface; I do not see how the general case follows from that.)

So, whom should we credit the result to? What would be a proper reference?

Intersection Theory. It might be worthwhile to take a look at theNotes and Referencessection of chapter 8, there are named many people who worked on versions of Bezout's theorem. $\endgroup$1more comment