It depends on the context whether M-theory refers to the full UV completion or just the 11D supergravity limit. For example when people say "type IIA string theory is related to M-theory by T-duality" and go on to calculate something in "M-theory", they really mean the supergravity limit. However, when you hear vague statements like "M-theory is the framework in which all known string theories are unified", then usually the full theory is meant, not just the supergravity limit.
When talking about p-branes, one usually means objects in string theory, extending along p spatial dimensions. As one may reduce a string theory to its supergravity limit, these objects live on as solutions of supergravity. A famous example is used in the original formulation of the AdS/CFT correspondence, where a stack of $D3$ branes (a special case of p-branes) actually corresponds to an $AdS_5\times S^5$ solution of classical (super)gravity (in a certain limit). In supergravity, black holes also find a generalization: so-called black $p$-branes are translationally invariant in $p$ dimensions, and form a black hole in the remaining ones.