I am considering over a field $k$ which is not algebraically closed, characteristic 0, and perhaps contains all the complex roots of unity that may appear. Feel free to realize it as some function fields such as $\mathbb{C}(x,y)$.

Given $X$ a del Pezzo surface of degree 2 (or any degree if there's a more general theorem), it is known that $X$ may be $k$-birational to a conic bundle with 6 (or 8-deg) degenerate fibers. It is also known that if $X$ is indeed a conic bundle then the Picard group has rank $\geq 2$. Do we have an iff theorem for determining a given del Pezzo surface is indeed a conic bundle? By "given" I mean for example an explicit equation $w^2=f_4(u,v,t)$.

I would also appreciate some explicit examples on this. I know Swinnerton-Dyer has studied conic bundles $r^2+s^2=f_2(t)f_4(t)$, but I don't think he gave example of del Pezzo surfaces birational to this form (I think he also worked over $\mathbb{Q}$ in stead of a function field).