Terminology: Variant of Density Let $X$ be a topological space.
Typically, when describing density, one says that if $Z\subseteq Y \subseteq X$ and the closure of Z in $X$, denoted $Y\subseteq cl(Z)$ then we say Z is dense in Y.
However, suppose that $cl(Y)\subseteq cl(Z)$, $Z \cap Y\neq \emptyset$ (but $Z\not\subseteq Y$ necessarily) then do we still say $Z$ is dense in $Y$, or does this have another term?
 A: 
Typically, when describing density, one says that if $Z\subseteq Y \subseteq X$ and the closure of Z in $X$, denoted $cl(Z)$, contains $cl(Y)$ then we say Z is dense in Y.

That's not how you typically define density. Usually a subspace $A$ of $X$ is dense in $X$ if $cl(A)=X$ (closure taken in $X$). Of course your condition is equivalent to that one. But the intermediate subspace looks like an unnecessary noise. 
Another thing is that since $Z\subseteq Y$ then $cl(Z)\subseteq cl(Y)$. Meaning your condition is actually $cl(Z)=cl(Y)$. This will be important in my next paragraph.

However, suppose that $cl(Y)\subseteq cl(Z)$, $Z \cap Y\neq \emptyset$ (but $Z\not\subseteq Y$ necessarily) then do we still say $Z$ is dense in $Y$, or does this have another term?

Most definitely we do not call that density. That condition is very weak, for example any space $Z$ with any subspace $Y$ will satisfy this (regardless of what the closure of $Y$ is). Unless (analogously to my previous paragraph) you actually mean $cl(Y)=cl(Z)$. This condition is a lot closer to density. But even in this case you would simply say that $Z$ and $Y$ have equal closures.
