The proof of Noether that the automorphisms of $\mathbb{C}(x,y)$ are generated by $\mathrm{PGL}(3,\mathbb{C})$ and the standard quadratic transformation had troubles because of infinitely near points. The proof of Castelnuovo (1907) and is completely valid and is the first one which is accurate. This is why the theorem is Called "Noether-Castelnuovo theorem". You can find a lot of proofs of this result. See for instance the book "Geometry of the Plane Cremona Maps" of Maria Alberich Carramiñana, which gives a detailed proof and explains the history of thre problem.

As YCor said, knowing the generators does not directly give a good understanding of the group. For instance, it is useless here to find conjugacy classes of elements of finite order (but the classification is known using birational geometry). There are many other results on the group that you can find in the literature.

The proof of Noether that the automorphisms of $\mathbb{C}(x,y)$ are generated by $\mathrm{PGL}(3,\mathbb{C})$ and the standard quadratic transformation had troubles because of infinitely near points. The proof of Castelnuovo (1907 is completely valid and is the first one which is accurate. This is why the theorem is Called "Noether-Castelnuovo theorem". You can find a lot of proofs of this result. See for instance the book "Geometry of the Plane Cremona Maps" of Maria Alberich Carramiñana, which gives a detailed proof and explains the history of thre problem.

As YCor said, knowing the generators does not directly give a good understanding of the group. For instance, it is useless here to find conjugacy classes of elements of finite order (but the classification is known using birational geometry). There are many other results on the group that you can find in the literature.