As you explain, the classification of birational involutions of $\mathbb{P}^2$ given by Bayle and Beauville in the article you cite gives the answer.

The Jonquières involutions preserve a rational fibration. They are in fact conjugate to $(x,y)\mapsto (x,p(x)/y)$ for some polynomial $p\in \mathbb{C}$ without any multiple root and of even degree. The fixed locus is given by the curve $y^2=p(x)$, which is rational if $\deg(p)=2$, elliptic is $\deg(p)=4$ and hyperelliptic of genus $g$ when $\deg(p)=2g+2\ge 6$. The birational type of the curve determines the conjugacy class in the Cremona group $\mathrm{Aut}_{\mathbb{C}}(\mathbb{C}(x,y))$. In particular, for $\deg(p)=2$ you have only one conjugacy class, in fact also conjugate to $(x,y)\mapsto (x,-y)$ in $\mathrm{Aut}_{\mathbb{C}}(\mathbb{C}(x,y))$ (but not by an element that fixes $x$).

The other involution (Geiser and Bertini) do not preserve any rational fibration. They preserve elliptic fibrations, but this does not help so much to write explicitely the map. This can be done in practice but the formula are quite horrible. In particular, the degree of a Bertini involution is $17$ and the degree of a Geiser Bertini involution is $8$. So $x$ and $y$ are sent onto rational functions, quotients of polynomials of degree $17$ or $8$. Choosing the coordinates such that the point $[1:0:0]$ or $[0:1:0]$ (pencils of $x/z$ and $y/z$ corresponding to $x=cst$ and $y=cst$) you obtain polynomials of degree slightly less ($11$ and $5$) but it not so good for expliciting the maps. The geometric description given by Del Pezzo surfaces of degree $1$ and $2$ that you can find in the article of Bayle and Beauville is definitely better, I think.

As you explain, the classification of birational involutions of $\mathbb{P}^2$ given by Bayle and Beauville in the article you cite gives the answer.

The Jonquières involutions preserve a rational fibration. They are in fact conjugate to $(x,y)\mapsto (x,p(x)/y)$ for some polynomial $p\in \mathbb{C}$ without any multiple root and of even degree. The fixed locus is given by the curve $y^2=p(x)$, which is rational if $\deg(p)=2$, elliptic is $\deg(p)=4$ and hyperelliptic of genus $g$ when $\deg(p)=2g+2\ge 6$. The birational type of the curve determines the conjugacy class in the Cremona group $\mathrm{Aut}_{\mathbb{C}}(\mathbb{C}(x,y))$. In particular, for $\deg(p)=2$ you have only one conjugacy class, in fact also conjugate to $(x,y)\mapsto (x,-y)$ in $\mathrm{Aut}_{\mathbb{C}}(\mathbb{C}(x,y))$ (but not by an element that fixes $x$).

The other involutions (Geiser and Bertini) do not preserve any rational fibration. They preserve elliptic fibrations, but this does not help so much to write explicitely the map. This can be done in practice but the formulas are quite horrible. In particular, the degree of a Bertini involution is $17$ and the degree of a Geiser Bertini involution is $8$. So $x$ and $y$ are sent onto rational functions, quotients of polynomials of degree $17$ or $8$. Choosing the coordinates such that the point $[1:0:0]$ or $[0:1:0]$ (pencils of $x/z$ and $y/z$ corresponding to $x=cst$ and $y=cst$) you obtain polynomials of degree slightly less ($11$ and $5$) but it not so good for expliciting the maps. The geometric description given by Del Pezzo surfaces of degree $1$ and $2$ that you can find in the article of Bayle and Beauville is definitely better, I think.