"Quasi-static" = "reversible" if you push the meaning of quasi-static far enough to get a universally meaningful definition. Somehow, when physicists invented the word quasi-static for classical thermodynamical experiments, I guess they actually meant what we define today as reversible. The old definition (slow) was too informal and did not take into account all the subtle processes like friction and viscosity.

The word quasi-static has fallen out of flavor since, and is mainly mentionned as a curiosity. Think of the [Joule Thompson expansion:][1]. No text really cares saying if it is "quasi-static" or not in an old fashion sense. What matters is that it is irreversible. Hence the confusion in most text books. It is not very useful to give a precise definition for quasi-static, because if you tried seriously, the only definition could get in the end, would be the one of "reversible".

In statistical mechanics (based on classical mechanics), a truly "quasi-static" process could be defined as:

"The process is equivalent to a progressive change in the system's Hamiltonian that is:

- known and controlled, independant of the unknown micro-state (position in the phase space)
- **slow and smooth** enough so that the system has time to go through its full energy orbit (in the phase space) for each infinitesimal change in the Hamiltonian. In other words the system can be considered at equilibrium at each infinitesimal step."

This definition is officially the definition of "adiabatic reversible". When you write $dU = -PdV$ ($V$ is any variable the Hamiltonian depends on and $P$ is the generalized force), you mean this kind of process.

Usual examples : 

- moving the piston of an insulated gas chamber (much slower than the speed of sound)
- moving a piston of a gas chamber in thermal contact with another gas (slow enough to allow thermal equilibrium at each step). In this example, the system under consideration is the union of the two gases. 
- Counter-example : the irreversible [Joule (free) expansion][2]

This definition excludes heat, since in the case of heat, the Hamiltonian varies in a way that is dependant on the micro-state (during each collision with a mollecule from the other system for example). It excludes friction. If the motion happens to be very slow but the pores are very small, then the Hamiltonian varies abruptly. It also excludes this interesting case mentioned by Huang : “a gas that freely expands into successive infinitesimal volume elements”. Indeed, if the potential wall is smooth, this cannot be a free expansion but a usual isentropic expansion.


  [1]: https://en.wikipedia.org/wiki/Joule%E2%80%93Thomson_effect
  [2]: https://en.wikipedia.org/wiki/Joule_expansion