> Strictly, realizable paths are an extremum (maximum, minimum, or inflection point) of the action.

At a realizable path, the action can be a local minimum or a saddle point but it's never a local maximum.

Tangentially, while "principle of least action" is a misnomer, it's not too severe of a misnomer, because realizable paths always minimize the action on a short enough time scale.

A reference for both these claims: http://www.eftaylor.com/pub/Gray&TaylorAJP.pdf

(I haven't read it to be honest but the abstract confirms what I'm saying)