What is a minimum work transition in stochastic thermodynamics?

We reassess the concept of transition at minimum work in classical stochastic finite-time thermodynamics, when the system dynamics is modelled by a diffusion process. We show that a well-posed formulation of the optimal control problem corresponding to the minimization of the mean work done on the system during a finite-time transition necessarily requires taking into account speed limits on control protocols. This fact has major qualitative consequences. First, it permits to discriminate between optimal swift engineered equilibration and transitions at minimum work. Second, it shows that in the limit when speed limits are removed, only transitions specified by generalized Schr\"odinger bridges admit a consistent physical interpretation. To illustrate these points, we focus on the simplest model problem: a levitating particle in a Gaussian moving trap.