I found a definition of the quadratic Hilbert Symbol (the one with $(a,b)=1$ if $ax^2+by^2=z^2$ for some $(x,y,z)≠(0,0,0)$ ) for example here or here.
I am now looking for a proof that this follows from the original definition of this Symbol by Hilbert.
We can simply rewrite Serre's definition $ax^2+by^2=z^2$ as $$ ax^2=z^2-by^2={\rm Norm}_{K/k}(z+y\sqrt{b}), $$ where $K=k(\sqrt{b})$ is now a quadratic field extension of $k$. This is exactly how the linked German text defines it.