Haste Additional Attack – Can It Be Used to Shove a Creature in DnD 5E?

The restrictions for using the Feat are clear:

• You must have taken the Attack action.
• You must have attacked only with glaive/halberd/quarterstaff.

The first bullet is easy-go: yes, you took the Attack action. The second bullet is probably a no, since you have attacked (as Shove is "a special melee attack") with something else that does not involve your quarterstaff (per Crawford below).

In a similar question, which asked

Does Shove qualify you to use the bonus attack in Two Weapon Fighting or Martial Arts?

Crawford answered

The shove and grapple options don't involve an attack with a weapon or an unarmed strike, so no. ¹

This means that both your interpretations in the "only" part lead to the same thing.

it's unclear if it mean you can't attack with other weapons, or that you can't take any actions other than polearm attacks

You are both "attacking with other weapons (where the other weapon is, probably, your body)" and "taking other (than polearm attacks) action".

¹ My interpretation from that is that No, you don't use your quarterstaff for the shove action. The reasoning for that is that if it does not involve an attack with a weapon, it either does not involve an attack (which is false, since it does) or does not involve a weapon (which is the only possible solution). It doesn't require a free hand because you can use your body, you don't need to push the person with your hands.

Haste grants an additional action (but limits the use of it)

Haste says says (emphasis added):

Until the spell ends, the target's speed is doubled, it gains a +2 bonus to AC, it has advantage on Dexterity saving throws, and it gains additional action on each of its turns. That action can be used only to take the Attack (one weapon attack only), Dash, Disengage, Hide, or Use an Object action.

The fact that you get an "additional" action indicates that this action is no different than your "normal" action, except for the limitations described in the next sentence. Nothing in those limitations turns the action into anything other than an action. Hence, it is an action, and it is yours, so it is your action, just as much as your normal action is. Note that Haste uses the same wording as the fighter's Action Surge, which unambiguously grants you a 2nd action that is indistinguishable from your first action:

On your turn, you can take one additional action.

As for how the additional action granted by Haste interacts with the other effects you've mentioned:

Witch Bolt

A literal reading of the spell suggests that yes, using the action granted by Haste to do anything would cause Witch Bolt to end. You could justify this logically by saying that maintaining Witch Bolt requires nearly all of your attention, leaving only your bonus action to do something else. However, Rykara's answer to the question that you linked makes the point that the rules are typically written for the common case of only 1 action without regard for exceptions. As such, it wouldn't be unreasonable for a DM to rule that Witch Bolt is maintained as long as you spend one of your actions on each turn to maintain it, allowing you to take other actions while maintaining the spell.

Charger

If you take the Dash action using the action granted by Haste, you have used your action to dash, which means you have satisfied the requirements to use your bonus action to make an attack as described in the feat. It doesn't matter which action you used to dash.

Turn Undead vs Fear

As described in the linked question, Turn Undead sets limits on what actions an affected creature may take, only allowing them to Dash, attempt to escape from something, and sometimes Dodge. In contrast, a literal reading of Fear just says that you must take one Dash action on each of your turns while frightened. It doesn't say that you can't do anything else, so once you've taken the Dash action and used all your movement to flee from the caster, you have done everything the spell requires you to do, and you are free to use the rest of your turn, even if the rest of your turn includes one or more actions (although you are still limited by the frightened condition).

Of course, for a creature that has only one action, the two wordings are effectively equivalent, so it feels a bit unsatisfying that granting an additional action causes them to diverge, especially in light of the observation mentioned above that most rules are only written for the case of having a single action. As DM, you can either accept that different effects sometimes work differently in certain situations for no logical reason, or you can house rule it by choosing one or the other interpretation and applying it to all similar "forced to flee" effects: either you are forced to take one Dash action, or all your actions must used to dash.

Having multiple actions makes the rules ambiguous

Ultimately, you are correct that the rules are a bit under-specified and ambiguous in cases where a creature can take multiple actions in a turn. "Your action" is unambiguous when you have one action, but once you gain multiple actions, nothing in the rules specifies whether "your action" should generalize to "all of your actions" or "one of your actions". In some cases, a literal reading of a specific effect appears to prefer one or the other interpretation, but in general there is no way to solve this unambiguously. 5th edition intentionally avoids using the rules to spell out the resolution of every weird edge case. Instead, it delegates the task of resolving those edge cases to the DM.