To answer your question directly, your action is your turn.
As you pointed out, in the Basic Rules, it says:
On your turn, you can move a distance up to your speed and take one action. You decide whether to move first or take your action first. Your speed--sometimes called your walking speed--is noted on your character sheet.
The most common actions you can take are described in the Actions in Combat section. Many class features and other abilities provide additional options for your action.
The list of Actions in Combat: Attack, Cast a spell, Dash, Disengage, Dodge, Help, Hide, Ready, Search, and Use an object. It does mention class features and abilities can give a character more choices, but for that, you would need to find something specific to a class.
The paragraph also says, "most common". This is because it is not an inclusive list. There will be actions a character can take that don't fall into one of the listed headings. An example is talking. It does not appear on the list, but it is definitely something a character can do. It's up to the DM to decide if it requires a full action or can be part of something else.
Reaction is the term for reacting to a trigger. However, it does not appear on the list of possibilities of an Action.
Certain special abilities, spells, and situations allow you to take a special action called a reaction. A reaction is an instant response to a trigger of some kind, which can occur on your turn or on someone else's. The opportunity attack, described later in this section, is the most common type of reaction.
Reactions only happen due to special abilities, spells, and situation. Since it is a named course of action (lower case), like Bonus Action, I would consider it not available during an Action.
There is an in-between
You can Ready an Action (appears on the list):
Sometimes you want to get the jump on a foe or wait for a particular circumstance before you act. To do so, you can take the Ready action on your turn, which lets you act using your reaction before the start of your next turn.
You use your Action to prepare for a trigger and use a Reaction. But that becomes your Action.
But Readying has its own problems. You have to declare what the trigger is and what you will do in response to that trigger before hand.
So Alberich could have said something like, "I'm going to walk towards Xanax, but if any of its minions start to attack me, I'll cast mage armor."
Now we have another problem. If your readied Action is casting a spell, you have to cast it as your Action. Then hold it, as if it was a Concentration spell, until the trigger happens.
When you ready a spell, you cast it as normal but hold its energy, which you release with your reaction when the trigger occurs. To be readied, a spell must have a casting time of 1 action, and holding onto the spell's magic requires concentration.
So in your scenario, Alberich has to have cast the mage armor spell to begin with. More over, if the trigger does not occur, the spell energy is not released so they have wasted the spell.
But Alberich shouldn't feel bad. Melee classes lose out when they use the Ready Action. Since the Attack Action doesn't happen on their turn, as pointed out in this question, where the character would normally get multiple attacks, their Readied Attack Action is reduced to a single attack (similar to an Attack of Opportunity).
All in all, Alberich is going to get hit. And if they survive, maybe they'll make a checklist.