Really what you're seeing in most of the above examples is that whenever you have a proper ideal $I$ extending the non-stationary ideal on $\kappa$, then ``not being in $I$'' gives you a notion intermediate between stationary and club. So realy we might ask instead for examples of natural proper ideals $I$ on $\kappa$ that extend the non-stationary ideal.

One family of such ideals I haven't seen mentioned yet arises from the theory of club-guessing. Concentrating on one particular case, suppose $\kappa$ is a regular cardinal greater than $\omega_1$, and let $S$ be the stationary subset of $\kappa$ consisting of ordinals of countable cofinality.

There is sequence $\langle C_\delta:\delta\in S\rangle$ such that each $C_\delta$ is club in $\delta$ of order-type $\omega$, and for every closed unbounded $E\subseteq\kappa$ there are stationarily many $\delta\in S$ with $C_\delta\setminus E$ finite. (So $C_\delta$ is almost contained in $E$ for stationarily many $\delta\in S$.)

There is a natural way to get a normal filter out of the above set-up: For $E\subseteq\kappa$ club, let $S_E=\{\delta\in S: C_\delta\subseteq^* E\}$. These sets generate a normal filter $F$ concentrating on the stationary set $S$, and the condition ``positive with respect to $F$'' gives you a nice intermediate notion.

Now look at the collection of $A\subseteq\kappa$ with the property that for $F$-almost all $\delta\in S$, $A\cap C_\delta$ is finite. (These are sets that ``run away'' from the sequence $\langle C_\delta:\delta\in S\rangle$.)

This collection generates a proper ideal $I$ on $\kappa$, and any $I$-positive set is also stationary: if $A\notin I$ and $E\subseteq\kappa$ is club, we can find a $\delta$ for which $C_\delta\subseteq^* E$ and $A\cap C_\delta$ is infinite, so $A\cap E\neq\emptyset$.