Can some one help me understanding why the Definition of Generalized Conic Bundle is generalization of the Conic Bundle definition.
This is the definition of a conic bundle from "Comparison theorems for Conic Bundle"- P. E. Newstead
And this is the definition of generalized Conic Bundle from "Cohomology of certain moduli space of vector bundles"- Vikraman Balaji
Well, the first definition says that we have a bundle $Y$ cut out of the trivial $\mathbb{P}^n$-bundle on $X$ -- that is, $X \times \mathbb{P}^n$ -- such that the fiber over each point of $x$ is a conic and such that $Y$ is locally a quadric surface in some three-dimensional linear subspace of the $\mathbb{P}^n$ bundle.
The second definition replaces the trivial $\mathbb{P}^n$-bundle with an arbitrary $\mathbb{P}^2$-bundle, and imposes the same conditions -- each fiber is a conic, and the family is locally cut out of $U \times \mathbb{P}^2$ by homogeneous quadratic polynomials. This generalizes the first definition because we can take our $\mathbb{P}^2$-bundle to be the one cut out of the trivial $\mathbb{P}^n$-bundle by the equations (1).