Can multiple Legendary Actions be used in 1 round?

Question

I had a whole thing typed for this and then saw I need to broaden the scope of my question.

If a creature has 4 legendary actions on its sheet (like the lich) and it states:

The lich can take 3 legendary actions, choosing from the options below. Only one legendary action can be used at a time...

Does that mean that per round of initiative - The lich can only use only 1 legendary action from the list up to a total point value of 3 (after multiple players turns)?

or

Does that mean that per round of initiative - the lich can use multiple legendary actions up to a total point value of 3 (after multiple players turns)?

Or

Have I completely missed the mark on both of these examples?

My confusion stems from the line:

Only one legendary action option can be used at a time.

Answer 1

Yes, the creature can take as many as listed, provided there are enough combatants to give it turns to do so

Legendary actions are explained on p. 11 of the MM:

A legendary creature can take a certain number of special actions- called legendary actions- outside its turn. Only one legendary action option can be used at a time and only at the end of another creature's turn. A legendary creature regains spent legendary actions at the start of its turn. It isn't required to use its legendary actions, and it can't use legendary actions while incapacitated.

So the lich can take up to 3 legendary actions in one round, if there are enough other combatants to provide three other turns, during each of which it can take one. If there is only one opponent, it could only take one legendary action, at the end of the opponent's turn, because it can only take one each time.

The lich's legendary options that are listed are:

Cantrip. The lich casts a cantrip.
Paralyzing Touch (Costs 2 Actions). (...)
Frightening Gaze (Costs 2 Actions). (...)
Disrupt Life (Costs 3 Actions). (...)

Each one is a legendary action, but some of them cost the lich more than one of its legendary actions when it is taking that one.

Examples

For example, if the lich decides on the end of the first opponent's turn to use its Frightening Gaze, that legendary action will consume two of the lich's actions for that round. It only will have one unused legendary action left (and since all of them except for the cantrip action cost more than one, will be reduced to using the cantrip option for the remainder of the round). It then could use the remaining legendary action to cast a cantrip at the end of the second opponent's turn. After that, it would have used up all its available actions for the round, and could not take another one at the end of the third opponent's turn.

Second example: if the lich had only one opponent, to not let go anything to waste it could choose to take the most expensive (and probably, powerful) option, Disrupt Life, at the end of that single opponent's turn, which will cost it all three of its available actions, so in taking the one action it is using all three potential actions.

Note that the other combatants do not need to be allies, so a cunning lich could keep some minions around, if it wanted to be able to take multiple legendary actions against a single opponent. (Kudos to @Dughall for pointing this out.)

Answer 2

The lich can take "legendary actions" while it is not the lich's turn to act.

After each other player acts, the lich has one opportunity to take a legendary action (any one of the four things listed).

The lich can take up to three total legendary actions in this way between the start of the lich's turn and the next start of the lich's turn.

Some legendary actions count as multiple actions towards this limit of three.

See here to read the same thing more officially.