thermodynamics - Reversible vs Quasistatic

I have some confusion about the definitions of a quasi-static and a reversible thermodynamic transformation.

As far as I understand, a quasi-static process is one that happens slowly enough for the system to remain in internal equilibrium at all times.

Example:

Consider a cylinder containing some gas in which we push a piston slowly to compress the gas. If the compression is sufficiently slow, the gas molecules have time to react and reach an equilibrium state inside the cylinder, so at any given moment the pressure and temperature of the gas are well defined. That is, the force per unit area is homogeneous across the walls of the container, and the average kinetic energy of any subsystem of gas particles is the same.

Non-example:

If on the other hand we pull up the piston more quickly than the fastest molecules in the gas, there will be a momentary vacuum separating the gas and the piston, which lasts until the gas will rush up to fill the entire volume of the whole space. During this adjustment, the pressure on the lower part of the cylinder is higher than the pressure on the piston (which is zero until the gas molecules reach it), so the pressure isn't well defined.

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Concerning reversibility, I have never seen a rigorous definition. Let me try to formalize what I have understood as best as possible. Let's call a path a function mapping a point in time to a point on the pV diagram: $p(t)=(V(t),P(t))$, defined on some interval of time $t \in [0,t_1]$.

We say a path $p$ connecting a state $A(V_0,P_0)$ to a state $B(V_1,P_1)$ is reversible if $p$ can be traversed in reverse, i.e if the path $p(t_1-t)$ for $t \in [0,t_1]$ is physically possible.

Intuitively, if we take a video of some process transforming a gas from state $A$ to state $B$ and rewind it, then it is possible to do an experiment which brings the gas from state $B$ to state $A$ which will be indistinguishable from the rewinded video.

Question 1: Is this a good definition of reversibility?

Question 2: Furthermore, I don't understand why Wikipedia says "Reversibility refers to performing a reaction continuously at equilibrium" - is this not the definition of quasistatic? Or do they mean a reaction continuously at equilibrium with its surroundings?

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Clear example of an irreversible process

The Joule-Guy-Lussac experiment, in which a barrier enclosing some gas in a subcompartment of a container is suddenly removed, allowing the gas to fill the whole volume of the container. Assuming the walls of the container and the barrier are adiabatic, this is not reversible because the internal energy of the system remains the same throughout the process, but work would be required to compress the gas back to its initial volume. Alternatively, if we took the gas in the state where it has filled the whole volume, and suddenly replaced the barrier, the gas would of course not go back its original state.

Also note that the above example is not quasi-static, since the barrier is removed suddenly.