Ambiguity: doubtfulness or uncertainty of meaning or intention; an unclear, indefinite, or equivocal word, expression, meaning, etc. A situation or statement that is unclear because it can be understood in more than one way.
"That Greek word's ambiguity has to make us take a step back and figure out just what kind of offerings Cain and Abel were okayed to bring that day."
It appears the ambiguity is on your part & to say 'we have to take a step back' is going beyond your ability to verify that this uncertainty of meaning or intention or doubtfulness is what the rest of us believe, which on my part is not true.
God says what He means & means what He says. 'God is not the author of confusion but of peace.'
You have pulled 3 verses OUT of their contexts, isolating them & them lumping them together because one word is common to all three verses. One cannot ignore the context from which they are taken & what is the intent of that verse in that context. Also one must observe the qualifying phrases within each verse that determine how that word is to be understood.
Also one must harmonize all Scripture that is pertinent to the topic of this thread, i.e., contrasting the sacrifice of Cain vs the sacrifice of Abel & why God accepted one & not the other.
Genesis 2:2-16 Now Abel was a keeper of sheep & Cain a worker of the ground.
In the course of time Cain brought to the Lord an offering of the fruit of the ground & Abel also brought of the firstborn of his flock & of their best (choice) portions. And the Lord had favor for Abel & his offering, but for Cain & his offering He had no regard.
So Cain was very angry & his face fell. The Lord said to Cain, “Why are you angry & why has your face fallen? If you do WELL, will you not be ACCEPTED? And if you do not do well, sin is crouching at the door. Its desire is contrary to you, but you must rule over it.”
8Cain spoke to Abel his brother. And when they were in the field, Cain rose up against his brother Abel & killed him. Then the Lord said to Cain, “Where is Abel your brother?” He said, “I do not know; am I my brother’s KEEPER?” And the Lord said, “What have you done?
The voice of your brother’s blood is crying to me from the ground. And now you are cursed from the ground, which has opened its mouth to receive your brother’s blood from your hand. When you work the ground, it shall no longer yield to you its strength. You shall be a fugitive and a wanderer on the earth.”
13Cain said to the Lord, “My punishment is greater than I can bear. Behold, you have driven me today away from the ground & from your face I shall be hidden. I shall be a fugitive & a wanderer on the earth & whoever finds me will kill me.”
Then the Lord said to him, “Not so! If anyone kills Cain, vengeance shall be taken on him sevenfold.” And the Lord put a mark on Cain, lest any who found him should attack him. Then Cain went away from the presence of the Lord & settled in the land of Nod, east of Eden.
I John 3:9-15 Anyone born of God refuses to practice sin, because God’s seed abides in him; he cannot go on sinning, because he has been born of God. By this the children of God are distinguished from the children of the devil: Anyone who does not practice righteousness is not of God, nor is anyone who does NOT love his brother.
11This is the message you have heard from the beginning: We should love one another. Do not be like Cain, who belonged to the evil one & murdered his brother. And WHY did he slay him? Because his own deeds were EVIL, while those of his brother were RIGHTEOUS. So do not be surprised, brothers, if the world hates you.
14We know that we have passed from death to life, because we love our brothers. The one who does not love remains in death. Everyone who hates his brother is a murderer & you know that eternal life does not reside in a murderer.
Jude 1:11 Woe to them! They have traveled the path of Cain; they have rushed headlong into the error of Balaam; they have perished in Korah's rebellion.
Heb 11:4 BY FAITH Abel offered God a BETTER sacrifice than Cain did. BY FAITH he was commended AS righteous WHEN God gave approval to his gifts. And BY FAITH he still speaks, even though he is dead.
Prov 29:10 Men of bloodshed hate a blameless man, but the upright care for his life.
Luke 11:48-51 So you are witnesses who consent to the deeds of your fathers: They killed the prophets & you build their tombs. "For this reason also the wisdom of God said, 'I will send to them prophets & apostles & some of them they will kill & some they will persecute. As a result, this generation will be charged with the blood of all the prophets that has been shed since the foundation of the world, from the blood of Abel to the blood of Zechariah.
Heb 12:24 And to Jesus the mediator of the new covenant, and to the blood of sprinkling, that speaketh better things than
that of Abel.
Matt 23:34-36 Because of this, I am sending you prophets and wise men & scribes. Some of them you will kill & crucify & others you will flog in your synagogues & persecute in town after town. And so upon you will come all the righteous blood shed on earth, from the blood of righteous Abel to the blood of Zechariah son of Berekiah, whom you murdered between the temple & the altar.
The sacrifices of Cain & Abel were physical sacrifices, but Abel offered his BY FAITH, that this type of sacrifice would please God, a blood sacrifice with the choicest of portions, for God alone, that spoke of Abel's righteous actions.
Cain offered his, seeking recognition for himself & what he had achieved & not by faith & to please God. He was extremely angry when his offering was rejected & thus he hated his brother to the point of murdering him. He was the first born and should have received the better portion.
This shedding of Abel's innocent blood is so important to God, it is repeated many times in Scripture. And Christ's eternal once for all sacrifice speaks even better than the blood of righteous Abel.