Thanks for all the comments and references!

I have updated the question an added a little bit of notation (and yes, Jung doesn't exactly write the polynomial automorphisms (or, "ganze birationale transformationen") in this form; he splits them into three basic ones). The question is indeed about all $k$-algebra automorphisms of $k(x,y)$. A reference to the statement for algebraically closed (and char 0) fields, indicated by @YCor, would be nice. Thanks!

Thanks, I'll think about it. If you develop your comment just a little bit in an answer, I can vote it up!

Does it matter that Bertini's theorem is formulated for projective varieties, when I'm interested in affine varieties?

Thanks! Perhaps you could elaborate a bit more on it, and write it as an answer? What about fields that are not algebraically closed?

I edited the question to include the assumptions that A is an algebra over the complex numbers and that the number of generators are finite, but the algebra is not necessarily finite dimensional.