Was the sin of Adam and Eve eating the forbidden fruit or believing in the lie of the serpent?

That is a good question. To answer that question lets look in the bible. When eve was talking to satan she told him that Jehovah God said not to eat the fruit of the tree in the middle of the garden. He then said,

"At this the serpent said to the woman: “You certainly will not die. 5 For God knows that in the very day you eat from it, your eyes will be opened and you will be like God, knowing good and bad." (Genesis 3:4, 5)

As you can gather from this scripture Satan didn't even tell Adam and Eve to eat the fruit. All he said is that they will not die if they do eat the fruit. Therefore we can't say believing the lie was the original sin because Satan didn't not tell them to do anything God told them not to. But, they did however eat the fruit which was directly against God's rule for them and was the original sin.

Eating the forbidden fruit. 

The actual temptation, per se is not sinful, but the yielding to it is sinful. James 1:14-15

It was the lie of this jealous angel that lead them to rebel and betray Jehovah God. Genesis 3;Genesis 6 and 19 chapter;Revelation 12:7,9,12; 1John 5:19

They Listened to Satan​—With What Results?

Bad things began to happen.

The sin was disobedience.  

Eve was wholly seduced by the serpent/liar/backbiter/accuser...did you never ask why, if their sin was eating an apple, why then did theysew fig leaves and hide their pvt part instead of covering their mouths ?.....there was no apple tree......and much more than th saducees and phariesses to watch/listen to told you.....

Rebellion in their hearts.

Their sin was disobeying GOD.

How could either of those things be considered 'sin' if Adam and Eve didn't know the difference between right and wrong yet?

If the invisible power behind the serpent had not sinned by deceiving Eve, she would never have eaten the forbidden fruit. Deception was the initial sin, and it was the deceiver who ever so slightly twisted the words of God to persuade Eve to believe his lie, that partaking of what was forbidden would elevate the couple to a higher place, alongside God. The deceiver claimed that a certain type of knowledge would elevate them but God had said it would lead to death. Further, God had planted the tree of life in the centre of the garden, and it would appear that the deceiver did not want them partaking of THAT God-given source of eternal life.

This is about what will lead to eternal life, and what will lead to death. God has warned humanity that partaking of a certain type of knowledge will lead to death - it is not knowledge that comes from Him, but from the deceiver. As the tree of life makes another appearance in the last book of the Bible, we can see the symbolism continues, and that God will yet restore humanity to what was the original state - for those who believe Him, and not the deceiver. God has given us His written word, and the Word of God made flesh - Jesus Christ - to deal with death once and for all. Knowing God and Christ leads to eternal life - John 17:6 - but all who are tempted by esoteric knowledge from the deceiver sin.

The sin was disobeying god.