Projection from point on line on quintic del Pezzo surface

Let $$X\subset \mathbb{P}^5$$ be a quintic del Pezzo surface embedded anti-canonically and suppose $$X$$ is smooth. Suppose further we are given a line $$L\subset X$$. After a suitable change of variables we may assume that $$L$$ is given by $$x_2=\cdots = x_5=0$$ if we work with coordinates $$x_0,\dots, x_5$$ on $$\mathbb{P}^5$$. Pick a point $$p\in L$$ and consider the projection $$\varphi\colon X\setminus p \to \mathbb{P}^4$$. For example, if $$p=(1\colon 0 \colon 0\colon 0\colon 0\colon 0)$$ this map is given by $$(x_0\colon x_1\colon \cdots \colon x_5)\mapsto (x_1\colon x_2\colon \cdots \colon x_5)$$. The closure of the image of $$\varphi$$ is a quartic surface $$Y$$ since $$p$$ is a non-singular point of $$X$$.

My suspicion is that $$Y$$ is a del Pezzo surface of degree 4 with a singularity at $$(1\colon 0 \colon \cdots \colon 0)$$. By this I mean that the anti-canonical divisor of $$Y$$ is very ample and has self intersection number 4.

It follows for example from Corollary 24.5.2 in Manin's "Cubic surfaces" that $$Y$$ is again a del Pezzo surface. The projection looks to me related to blowing-up $$X$$ in $$p$$, which would yield a birational surface whose self intersection number is 4, but I am not sure about a precise relation.

Does anyone know a reference where this situation is studied? Any help would be greatly appreciated!

The map, indeed, blows up the point $$p$$ and then contract the strict transforms of all lines passing through $$p$$ (which are $$(-2)$$-curves). Therefore, $$Y$$ is a singular del Pezzo surface of degree $$4$$ with one or two nodes, depending on whether the point $$p$$ lies on one line of $$X$$ or on two lines.