What are names for different approaches to unintentional sin?

"A rose by any other name is still a rose" said a national bard. This applies here. People can call sin by whatever names they think appropriate, but there is a strange tendency for those who think they can contribute towards their own salvation to claim that the word 'sin' should not apply to them (as they are on the path to salvation). There is a notable system in their ranks of speaking about their "failings" or "shortcomings" but they never seem to speak of themselves as sinners. They seem inclined to speak about those who profess no faith in Christ as sinners. But not themselves, for they claim they never deliberately sin - they merely fail to hit the mark of perfection, or they fall short of God's standards.

However, "A rose by any other name is still a rose" - and in the Bible, sin is always sin, even though it is unintentional. Consider how God dealt with unintended killing of a fellow human - manslaughter - with his covenant people. Six cities of refuge were built throughout the nation for such a manslayer to run to, to prevent being killed by the next of kin of the one he'd unintentionally killed. If he made it to a city of refuge, he then had to stay within the city bounds until the death of the High Priest in Jerusalem, otherwise the next of kin could still kill him. See Numbers ch.35 & Joshua ch.20.

This amply shows how, in the Bible, even unintentional killing of a human is such a sin in God's eyes that the person could suffer the death penalty, unless they followed God's merciful provisions. That is God's approach to manslaughter - provision of a city of refuge. It recognizes the difference between that and deliberate murder, but life-blood has still been spilt, as God stated in Genesis 9:6, and it cannot be swept under the carpet.

There is another aspect to unintentional sin that your question raises. There are sins of omission - not doing something right, that one knows they should (and could) do. Choosing to hold back when one could do good entails a sinful choice.

If we stop for just a few minutes to consider our sins of omission, never mind our unintended sins of commission, we would be mortified and utterly ashamed of ourselves. That's because sin is sin is sin - in God's eyes. "For all have sinned and come short of the glory of God" (Romans 3:23). Paul went on to explain how "sin, taking occasion by the commandment, deceived me, and by it slew me... But sin, that it might appear sin, working death in me by that which is good, that sin by the commandment might become exceeding sinful" (Romans 7:11-13). He admitted that sin dwelt in him, so that even when he tried to do good, sin was still present (vs. 19).

Paul could only find deliverance from sin (all sin) by the grace of God. This means that a person who becomes a Christian is delivered from their slavery to sin, and that when they do sin (which is at odds with their new nature in Christ), they find in Jesus Christ, the righteous, their Advocate (1 John 2:1). But notice how that verse speaks only of sin - not degrees of sin, because sin is sin is sin. Those who know liberation from slavery to sin become far more sensitive to sin than ever they were before conversion, for even though they no longer desire to sin, they don't think that lets them off. They don't minimize unintentional sin as merely "missing the mark" or being "imperfections".

The Bible never calls sin anything less than what it is - an offense against God, whether intended or unintended. None of us can do better than to make a systematic study of sin in the Bible. We need to find out from the Bible what sin actually is, in God's eyes, which should lead to heartfelt repentance and then discovering the grace and peace of God, who freely forgives. Once he has cleansed our conscience, we will seek to keep it cleansed by living to please him, not ourselves, and that is a massive step towards understanding how to avoid unintentional sin. Then we will learn through experience that the one approach to all sin is always via faith in what Christ did to deal with all our sin - receiving God's free forgiveness.