What is the definition of the Potential of Mean Force (PMF) in physics jargon?

Independently of being part of the physicists or chemists community, PMF has not a unique definition. Some forms of PMF have been introduced many decades ago in different contexts and I would advice to cross check what is the exact definition used in a given context.

Whatever is its definition, the potential of mean force embodies a statistical average which depends on the kind of macroscopic state (equilibrium or out of equilibrium) and even for equilibrium states, for finite size systems it depends on the chosen ensemble. Therefore, use of a PMF obtained in an equilibrium ensemble for non-equilibrium calculations is not a fully consistent procedure, although it could be seen as an approximation.

Finally, the expression for the probability density $mathcal{P}(x)$ is perfectly valid. Its intuitive meaning is to sum the probability of all the states such that the function $x({bf q})$ is equal to $x$.