I just noticed there is a definition that is "missing" from order theory. Let $(X,\leqslant)$ be any type of (pre-) ordered set and $S\subseteq X$. Define the predicates ($I$ for "inclusion," $C$ for "comparability")
\begin{align} I_\text{weak}&=x\in X & C_\text{weak}&=\forall s\in S:x\leqslant s\implies s\leqslant x \\\\ I_\text{strong}&=x\in S & C_\text{strong}&=\forall s\in S:s\leqslant x. \end{align}
Then we have
\begin{align} I_\text{weak}\wedge C_\text{weak}&\Longleftrightarrow\;? \\\\ I_\text{weak}\wedge C_\text{strong}&\Longleftrightarrow x\text{ is an upper bound of }S\\\\ I_\text{strong}\wedge C_\text{weak}&\Longleftrightarrow x\text{ is maximal in }S\\\\ I_\text{strong}\wedge C_\text{strong}&\Longleftrightarrow x\text{ is a maximum/greatest element of }S. \end{align}
Is there a term for the first condition, and is it ever relevant or useful?