It's a common newcomer mistake to expect that anode and cathode denote a polarity in voltage. The words have nothing to do with voltage and everything to do with current.
Cathode is derived from late Latin cathodus meaning "way out". It is the terminal of the device from which conventional current goes out into the external circuit. Conventional current is opposed to the direction of electron flow, so it is also the terminal where electrons enter the device from the circuit (the cathode of a vacuum tube emits electrons into the vacuum).
In a battery, the direction of current is different during charge and discharge. During charging, the current is forced through the battery by the circuit, so it flows through the battery from the + terminal to the - terminal. This makes the - terminal the cathode during charging.
During discharging, the battery forces current through the external circuit, so current flows through the battery from the - to the + terminal. This makes the + terminal the cathode during discharging.
You asked about what happens "inside the battery". From the point of view of electrical theory it's simply a voltage source in series with a resistance, there is nothing else in there. The electrochemical reaction between the cell plates and the electrolyte is a separate topic.
From the point of view of chemistry you are talking a different language, with things like redox potentials, half-cell voltages, and reaction formulas. Unlike wires in an electrical circuit, electrons really do not flow at all in the electrolyte, only ions do. The electrolyte contains + charged cations and - charged anions. The cations flow towards the cathode, and the anions flow toward the anode. Electrons do not flow.
I would also disagree with those who say that "it doesn't matter what convention you use, it's entirely arbitrary." Words are symbols, so they have meaning entirely and solely by convention. The relationships between words describe reality, which is independent of any convention. So to invert the meaning of any word destroys all knowledge we have about reality.