Does there exist a star-Lindelöf space which is not star-$L$-Lindelöf?

Question

1. A space $X$ is said to be star-Lindelöf if for every open cover $\mathcal U$ of $X$ there exists a countable subset $\mathcal V\subseteq\mathcal U$ such that $St(\cup\mathcal V,\mathcal U)=X$.

2. A space $X$ is said to be star-$L$-Lindelöf if for every open cover $\mathcal U$ of $X$ there exists a Lindelöf subset $Y$ of $X$ such that $St(Y,\mathcal U)=X$.

It is clear from the above definitions that every star-$L$-Lindelöf space is star-Lindelöf. But we don't know if there eixsts a star-Lindelöf space which is not star-$L$-Lindelöf.

Answer 1

The question you ask has already been answered by Song with different terminology in the following paper (see Example 2.3):

ON L-STARCOMPACT SPACES, Czechoslovak Mathematical Journal, 56 (131) (2006), 781–788

https://cmj.math.cas.cz/full/56/2/cmj56_2_40.pdf

In addition, you may also consider and check some properties of the star-P type in the following sense (defined in "Classes defined by stars and neighbourhood assignments" by van Mill et.al.):

Let P be a topological property. We said that a spaces X is star-P if for any open cover U of X, there exists a subspace Y of X with property P such that St(Y,U)=X.

Thus, what you call star-L-Lindelof is the same as star-Lindelof with above definition.