The Ten Commandments do define sin...
Explicitly covered by the Ten Commandments:
Pride
Greed
Gluttony
Sloth
Envy
Hatred
Witchcraft, Sorcery
Fornication
Discord, Dissensions, Factions
Wrath, Fits of Rage
Showing Favoritism, Prejudice and Discrimination
Selfish Ambition, Self-Centeredness
Atheism
Not sins:
Lust (beware of 1 Tim 4 teachers who teach that "sex is evil")
Jealousy (the 10 Commandments make clear that jealousy is one of the most important values to a Christian)
Dishonesty, Deception (these are the outworking of sin, not a sin in themselves)
Unbelief, Disbelief, Agnosticism, Indifference (lack of any belief is not a sin)
Withholding Remedy to Human or Animal Needs (not a sin in itself)
This leaves:
Impurity
Debauchery
Sodomy
Bestiality
Drunkenness
Drug Abuse
These are all symptoms of a rebellion against God and are broadly covered by the First Commandment.
For a spiritual man, the Ten Commandments affirm the spiritual truth within our hearts - they confirm what we know to be true and right, covering a broad range of sins, toward God and man. They are set apart in the Torah because they represent a summary of the whole Torah. And they are in order of importance.
Likewise, the Lord Jesus summarized the whole Torah and all 10 Commandments:
Luke 10:27
"You shall love the Lord your God with all your heart, with all your soul, with all your strength, and with all your mind, and your neighbor as yourself."
Now I'd like to mention the issue of...
Must Christians keep the 10 Commandments?
Absolutely not! Christians are not under the Torah - that was the Old Covenant which has been superseded. (Besides which, the Torah contains hundreds of laws, not just ten.) If you put yourself back under the Torah, then:
Galatians 5:4
"You are severed from Christ, you who would be justified by the law; you have fallen away from grace."
The Lord Jesus said the Torah is good, and it is good to observe it. Paul said the same. It is obvious and evident that everything the Lord commands is good, and leads to blessing.
But Jesus never said we must keep the Torah. That is legalism.
He said we are to keep HIS commandments under the New Covenant. The New Covenant is nothing at all like a Law or a set of regulations: it is a radical process of inner change, based upon faithfulness and obedience to Jesus Christ, and being spiritually transformed. This can only happen through complete and total submission to Jesus Christ and His commands, putting Him before all else, and striving to follow Him.
By the way, this is my first post here. I signed up to make this comment but not sure I will be staying as I quickly discovered that, as well as being highly commercial, this site has some very ungodly principles which I would not want to support.