Let $Y$ be a proper closed subspace of a normed space $X$ and let $x_0\in X\setminus Y$ be arbitrary.
When we speak of the subspace $Z\subseteq X$ spanned by $Y$ and $x_0$ what are you talking about? Are we talking about the subspace generated by the union? That
$$Z=\text{span}(Y,x_0)=\text{span}(Y\cup \{x_0\})=\text{span}(Y)+\text{span}(x_0)=Y+\text{span}\{x_0\}$$
It's correct?
Yes, that is the usual meaning of that phrase. To me the most direct "translation" is $\operatorname{span}(Y \cup \{x_0\})$. I assume you are familiar with the other equalities in your question (with the given assumptions on $Y$).