D&D 5E Casting multiple spells with bonus spells and the order they are cast.

[Caliban]

So yes, RAW would indicate that if you cast a spell as a bonus action, you can't cast a spell as a reaction if the reaction happens on your turn. But if the reaction happens on somebody else's turn, then you can cast a spell as a reaction.

Is that correct?

Yes, that is correct.

While in theory you could cast a cantrip as a reaction, the wording of the rule prevents that:
You can only cast a cantrip with a casting time of 1 action.
You can't cast a spell as a reaction (cantrip or otherwise) unless it has a casting time of 1 reaction.

Furthermore, no cantrips have a casting time of 1 reaction.

This is also true. (On one of my previous posts I think I said a PC would be able to cast a reaction cantrip if one existed, but I was incorrect.)

The barbarian stands in a doorway and Readies their action to attack the first creature that comes in reach. Then on the goblin's turn, the goblins close to attack. The barbarian's reaction is triggered and can attack, but cannot use the cleave ability of their feat because the description of that ability starts with "On your turn...."

Sure. You would not be able to kill/crit a creature as a reaction, and then take a bonus action attack on your next turn. The kill/crit to generate the bonus action attack has to take place on your turn.

That is, they are specifically excluding the cleave ability from a reaction - probably to eliminate it's use in an opportunity attack, but it has the same effect here.

So oddly, the cleave feature of GWF is the exact opposite of "you can't cast a spell as a reaction on your turn if you've already cast a spell as a bonus action."

You can only use the cleave ability of GWF on your reaction if your reaction occurs on your turn.
You can only cast a spell on your reaction after casting a spell with a bonus action if the reaction occurs on somebody else's turn.

Sure. Because the GWM feat doesn't care how you kill/crit something (i.e. using an action or a reaction) - only that it takes place on your turn. (Plus, you still have to have a bonus action available in order to make the extra attack.)

Would a 20th level fighter be able to use the Extra Attack feature for three attacks during their reaction if their reaction occurs on their turn?

In other words, could a fighter take 3 attacks for their action, then 3 more attacks with a second action using action surge, and 3 more attacks with their reaction, as long as the reaction occurred on their turn?

I'm not sure you can actually take the Attack action on a reaction, so I'm guessing no. But it ties into the example they used for Sneak Attack.

Depends on what the reaction is. An Opportunity Attack reaction specifically limits you to a single attack, so the fighter wouldn't get three attacks. If there was a theoretical ability that allowed you to take an Attack Action as a reaction (or even as a bonus action), then the fighter would be able to make all three attacks.

But currently no such ability exists - even Haste limits you to a single attack on the extra action it grants.