Do full charges and deep discharges wear the battery because of terminal voltage?
Indeed a discharge below 2.5 V will cause stress in many Li-Ion based batteries. Not all Li-Ion based batteries are the same though, some handle a deep discharge better than others.

But discharges below 2.5 V should be avoided and that's why all nearly all batteries used in electronic consumer devices have a battery protection circuit which will simply disconnect the battery when its voltage drops too low.

Usually the same circuit will also protect against overcharging and drawing too much current from the battery.

If a battery is shorted out in a way the voltage drops below 2.5V, does that have the same harmful effect?

1) You should NEVER short a charged Li-Ion based battery.

2) Obviously shorting the battery terminals makes the voltage across them equal to zero. Then the only resistance that limits the current flowing is the internal resistance of the battery. This internal resistance is quite low in most Li-Ion batteries. For sure the current that will flow will be larger than the maximum current that the battery can safely deliver. The current and voltage across this internal resistance means that this internal resistance needs to dissipate a lot of heat. And since the internal resistance is part of the battery's structure it "sits" inside the battery. So where does the heat go: inside the battery. So the battery will get hot. Heat is bad for batteries, even the engineers at Boeing underestimated that. What happens when the battery gets hot: more energy is released.

My point: you should not even be thinking about a shorted Li-Ion battery because that is outside their normal range of operation. Any condition outside the normal range of operation causes extra stress and thus a shorter battery lifetime.

Li-Ion batteries always wear out, it's like a continuous process. To slow this process down as much as possible it is recommended to (dis)charge the battery to about 40 % to 60 % charge level and then disconnect the battery (if possible).

The terminal voltage and battery voltage will be the same when no current is flowing. And that is the best condition to measure the battery voltage to estimate the charge level, when no current is flowing.

Your point about terminal voltage makes no sense. If the charging current causes the terminal voltage to be much higher than the battery's internal voltage then you're probably charging the battery too fast and you should change to a lower charging current.

Fast charging always causes more stress than slow charging but that has nothing to do with terminal voltage. Fast charging is always a compromise between how fast the battery is charged and the lifetime you want from the battery. Also fast charging ideally should only be allowed between charge levels of around 30%
to 80 %. Above and below that increased the stress on the battery.