No: actions in combat happen only within turns
"The Order of Combat" creates the structure of turns.
The 'Combat step-by-step sidebar' describes the structure of a combat:
3. Roll initiative. Everyone involved in the combat encounter rolls initiative, determining the order of combatants' turns.
4. Take turns. Each participant in the battle takes a turn in initiative order.
5. Begin the next round. When everyone involved in the combat has had a turn, the round ends. Repeat step 4 until the fighting stops.
This establishes that all actions within combat take place on someone's turn. In particular, there is no mention in (4) of a time that happens 'between the turns' of combatants, or in (5) of a time after the end of the last combatant's turn but before the new round and the start of the first combatant's next turn.
It could be argued that the rules for legendary actions, as specific over general, themselves establish these inter-turn times. To evaluate this claim, we should consider whether such an understanding would improve or worsen game play.
Even the author of the original claim that legendary actions occur between turns said that this is a 'bug in the rules', and that it doesn't make sense that Multiattack Defense would not be able to trigger off Legendary Actions. But Multiattack Defense is not the only 'for the rest of the turn' effect. Note that while many effects last 'until the end of the turn', or occur 'at the end of the turn', a few others last 'for the rest of the turn.' If all time occurs within turns, all of these effects occur over the same time period. But if 'the end of the turn' is actually after the turn ends, then 'for the rest of the turn' effects, like Multiattack Defense, will end before 'the end of the turn' (that is, before the supposed between-turn period). Thus Legendary Actions will not only fail to interact sensibly with Multiattack Defense, but with any such effects. For example:
Fangs of the Fire Snake. When you use the Attack action on your turn, you can spend 1 ki point to cause tendrils of flame to stretch out from your fists and feet.
Your reach with your unarmed strikes increases by 10 feet for that action, as well as the rest of the turn.
Suppose my monk used this ability on her turn, and then, at the end of her turn, a foe used a legendary action to leave her increased reach. She would not be able to respond with an opportunity attack because the movement at the end of her turn occurred after her turn.
Suppose my party's fighter, seeing the Legendary Foe slipping past my monk, tried to use his Sentinel Feat to stop it:
When you hit a creature with an opportunity attack, the creature’s speed becomes 0 for the rest of the turn.
If it was Legendary Movement, then it doesn't take place on a turn; rather it happens between turns, so the Feat would have no effect. Does your Rogue want to apply Sneak Attack bonus damage to their Opportunity Attack on the creature using Legendary Movement? They can't, because Sneak Attack can be used only once per turn and Legendary Actions don't take place on a turn...
Clearly this interpretation, that Legendary Actions take place between turns, creates complications with many other abilities and effects. I can't, however, think of anything that such an interpretation adds in compensation, no rules that become clearer as a result. This interpretation doesn't gain us anything.
What is the 'end of the turn' for?
What can you do on Your Turn?
On your turn, you can move a distance up to your speed and take one action...
Various class features, spells, and other abilities let you take an additional action on your turn called a bonus action...Your turn can include a variety of flourishes that require neither your action nor your move.
The bulk, or body, of your turn is spent resolving actions and effects that you elected. It is your turn because the principle components are your decisions about what to do.
However, there are other portions of your turn - the beginning of your turn, and the end of your turn. At the beginning or start of your turn are effects that happen before you have a chance to decide anything - they happen to you rather than them being your choice. Things like environmental damage and death saves occur at the start of your turn because their results can dramatically affect the options you have available once you reach the part of your turn where you can elect your own actions.
The end of your turn (which is still your turn), is for everything that resolves once you are done making decisions. Once you have exhausted all of your possible options, you 'pass', and begin applying end-of-turn effects. These are typically things like saves against debilitating effects and the end of the duration on enhancements. They happen after the elective portion of your turn is over specifically so that you cannot respond to any state-changes with new actions. When you make your save against hold person, you can't declare that now you would like to use your movement. You can't, because it is already the end of your turn.
For a creature's Legendary Actions to occur at the end of your turn indicates that you cannot respond to them with any of your main actions - that part of your turn is over. But it is still your turn, and you (and others) can still respond to them on your turn with things like reactions or other abilities that specifically allow you to respond, and with all of the benefits that it still being your turn, or someone's turn, confer. This is as it should be.
If you instead remove Legendary Actions to a time between turns, you unnecessarily invent a time that disrupts these interactions for no gain other than coming down on one side of what is ultimately a debate about language.
