Petri nets and their numerous extensions (or subclasses) are one of the popular traditional formalisms for the specification and verification of concurrent systems. Furthermore, due to the expressivity of rewriting logic, Maude and its associated analysis tools have been adopted in many recent works for executing and analyzing Petri nets. In this paper, we first present the existing semantics for the standard Petri nets. Then, we demonstrate the usefulness and the expressivity of the new enhanced semantics-that encodes the sets of tokens in a place by their cardinality-for such type of Petri nets. Thereafter, we show that such semantics can naturally express different variants of high-level Petri nets such as Petri nets with inhibitor arcs, variable arc weights, and coloured Petri nets. The distinguishing feature of this semantics is that it facilitates the checking of behavioral properties related to boundedness and liveness via the Maude LTL model checker.