Was the identity of Judas revealed more than once?

[MarieP]

I'm wondering if the identify of Judas as Jesus' betrayer didn't happen more than once that night in the upper room. Once during the meal before the Lord's Supper (Matthew/Mark), once after it (Luke), and once after the washing of the disciples' feet (John).

Matthew 26
21 Now as they were eating, He said, “Assuredly, I say to you, one of you will betray Me.”
22 And they were exceedingly sorrowful, and each of them began to say to Him, “Lord, is it I?”
23 He answered and said, “He who dipped his hand with Me in the dish will betray Me. 24 The Son of Man indeed goes just as it is written of Him, but woe to that man by whom the Son of Man is betrayed! It would have been good for that man if he had not been born.”
25 Then Judas, who was betraying Him, answered and said, “Rabbi, is it I?”
He said to him, “You have said it.”

Luke 22
21 But behold, the hand of My betrayer is with Me on the table. 22 And truly the Son of Man goes as it has been determined, but woe to that man by whom He is betrayed!”
23 Then they began to question among themselves, which of them it was who would do this thing.

John 13
21 When Jesus had said these things, He was troubled in spirit, and testified and said, “Most assuredly, I say to you, one of you will betray Me.” 22 Then the disciples looked at one another, perplexed about whom He spoke.
23 Now there was leaning on Jesus’ bosom one of His disciples, whom Jesus loved. 24 Simon Peter therefore motioned to him to ask who it was of whom He spoke.
25 Then, leaning back on Jesus’ breast, he said to Him, “Lord, who is it?” 26 Jesus answered, “It is he to whom I shall give a piece of bread when I have dipped it.” And having dipped the bread, He gave it to Judas Iscariot, the son of Simon. 27 Now after the piece of bread, Satan entered him. Then Jesus said to him, “What you do, do quickly.”...30 Having received the piece of bread, he then went out immediately. And it was night.

In one account, Judas was shown by dipping his hand along with Jesus (revealed after the fact- could this be washing, not the dipping of bread?). In the other, Jesus says He will give him the sop. These seem like two different events to me.

John 13:28-29 shows the disciples' slowness to learn, so what do you think?:

28 But no one at the table knew for what reason He said this to him. 29 For some thought, because Judas had the money box, that Jesus had said to him, “Buy those things we need for the feast,” or that he should give something to the poor.

I know there's discussion of whether or not Luke is putting the events in chronological order, but does he speak of things in non-chronological order anywhere else?