“This cup, poured out for you, is a new testament of my essence." (Luke 22:20 DT)

This verse is being quoted from the Devotional Translation. There is a noticeable difference between this translation and translations of this statement found in most sectarian versions of the Bible. For example, here is the translation from the King James Version:
"This cup is the new testament in my blood, which is shed for you."
Here is the translation from the New Living Bible (NLT):
“This cup is the new covenant between God and his people—an agreement confirmed with my blood, which is poured out as a sacrifice for you."
Here is the translation from the New International Version:
“This cup is the new covenant in my blood, which is poured out for you."
So it is all over the place. We can see that sectarian translators have inserted their interpretation of what Jesus is trying to say.

Furthermore, their various interpretative translations make little sense. “This cup is the new covenant in my blood?"  "This cup is the new testament in my blood?" “This cup is the new covenant between God and his people?" These statements make little sense. Plus they are quite grotesque.

The key word that has captured their attention is the Greek word αἷμα (haima), which most have translated to "blood."

As if Jesus is talking about his blood. Is he really asking his disciples to drink his blood?

That is quite the cannibalistic activity, yes? Why would Jesus want his students to drink his blood? Maybe the better question is why would these translators and teachers want to drink Jesus' blood?

Because they want to take advantage of Jesus' suffering. They revel in Jesus' persecution because they think that Jesus' suffering on the cross is cleansing their sins.

If Jesus' suffering on the cross cleanses everyone's sins, then why are so many people suffering the consequences of their sinful activities? Why are people going to jail or being put to death for murder?

If Jesus' dying on the cross saved everyone from sin then there would be no consequences for our sins. How would that work? Are they saying that we can just do whatever we want, and it doesn't matter, cause our sins would be cleansed by Jesus' persecution?

Why then did Jesus instruct this man whom he healed:
"Stop sinning or something worse may happen to you.” (John 5:14)
If Jesus' persecution on the cross was going to cleanse everyone of their sins, why would he instruct this man to stop sinning?

And why did Jesus' disciples ask this question about a man born blind:
“Rabbi, who sinned, this man or his parents, that he was born blind?” (John 9:2)
It is obvious from this question that Jesus' students understood that sinning in a previous life has consequences in the next life - since it was one of the possibilities explaining why the man was born blind.

We can thus know from these exchanges that Jesus taught the law of consequences. This teaching resulted in the famous clause, "as you sow, so shall you reap" - from Galatians 6:7.

If everyone's sins are cleansed by Jesus' dying on the cross, why didn't he and his followers teach, "as you sow, so shall I reap?"

So what is the meaning of the Greek word αἷμα (haima) and what did Jesus mean by this statement?

The Greek word αἷμα (haima) can certainly mean "blood" when the physical body is being discussed. But Jesus wasn't speaking of his physical body here. He was speaking of his spiritual self. He was speaking of the self that transcends the physical body. This is why Jesus said:
“Therefore I tell you, do not worry about your life, what you will eat; or about your body, what you will wear." (Luke 22:12)
The spirit-person does not circulate blood like the physical body does. The spirit-person circulates love. This is why each of us - from within - always seeks to love and be loved.

This is also Jesus' essence: Love. Jesus loved the Supreme Being and he wanted his disciples to also love God. This was Jesus' "new testament:" Loving the Supreme Being with all his heart.

This is Jesus' most important teaching:
"'Love the Lord your God with all your heart and with all your soul and with all your mind.' This is the first and greatest commandment." (Matt. 22:37-38)
This is Jesus' essence. This is why it is his "first and greatest" instruction.

So why did Jesus say the "cup" is being "poured out" for them? Consider Jesus' use of the "cup" from Jesus' prayer to God, which occurred later that evening:
“Father, if you are willing, take this cup from me; yet not my will, but Yours be done.” (Luke 22:42)
Here Jesus also uses the word "cup" - translated from the Greek word ποτήριον (potērion) - which can mean "cup" or "vessel" according to the lexicon. When used metaphorically, according to Thayer's lexicon, it can mean:
"one's lot or experience, whether joyous or adverse, divine appointments, whether favourable or unfavourable, are likened to a cup which God presents one to drink: so of prosperity and adversity"
The reason Jesus uses the word "cup" or "vessel" in both statements is because the "cup" or "vessel" represents his situation and position. Jesus was praying to God to take his "cup" from him because he wanted to give himself to the Supreme Being. He wanted to please God with his life.

He was lovingly devoted to the Supreme Being. He wanted his life to be about and for God. This makes this loving devotion to God Jesus' very essence - his very life.

Jesus was trying to give this loving devotion to his students, because he knew this would save them. He wanted them to also taste love for the Supreme Being. This is what Jesus wanted them to drink up - love for the Supreme Being - which was the essence of Jesus' teachings and thus his very essence as well.