The Lagrangian doesn't minimizes the action. The Lagrangian is the integrand of the integral defining the action. The action is a function of trajectories. The trajectories that make the action stationary (not necessarily minimal) are the physically realizable trajectories. The Lagrangian isn't a trajectory so it doesn't even make sense to ask whether it minimizes the action.

Really, it's hard to define "Lagrangian" on its own. Pretty much any function at all can be called a (potentially higher- or lower-order) Lagrangian. You can define an action from it and produce (potentially higher- or lower-order) Euler-Lagrange equations from it. "Lagrangian" is a term that refers to part of a larger system, namely Lagrangian mechanics, in the same way "integrand" refers to a part of a larger expression. You can't classify arbitrary functions into ones that are integrands and ones that aren't.