Is there a name for the idea that Jesus takes away the sin of the world by causing those who abide in him to sin less and be more righteous?

It seems you're talking about Justification, which more relates to making one less sinful and more holy, and "which denotes the transforming of the sinner from the state of unrighteousness to the state of holiness and sonship of God"

Atonement is connected with this, and is also how one understands how Christ "takes away the sins of the world."

"Hence, in Catholic theology, the Atonement is the Satisfaction of Christ, whereby God and the world are reconciled or made to be at one. "For God indeed was in Christ, reconciling the world to himself" (2 Corinthians 5:19)."

The Catholic doctrine on this subject is set forth in the sixth Session of the Council of Trent, chapter ii.

Whence it came to pass, that the Heavenly Father, the Father of mercies and the God of all comfort (2 Corinthians 1, 3), when that blessed fullness of the time was come (Galatians 4:4) sent unto men Jesus Christ, His own Son who had been, both before the Law and during the time of the Law, to many of the holy fathers announced and promised, that He might both redeem the Jews, who were under the Law and that the Gentiles who followed not after justice might attain to justice and that all men might receive the adoption of sons. Him God had proposed as a propitiator, through faith in His blood (Romans 3:25), for our sins, and not for our sins only, but also for those of the whole world (I John ii, 2).

There is a name for the process whereby those who abide in him are empowered to sin less. As John said (in 1 John 3:5-6) no-one who remains in him keeps on sinning. It would be impossible for anyone to accomplish this by themselves because we are all sinners. Yet Christ Jesus enables those who abide in him to sin less. That process is called sanctification.

Sanctification is the deliverance from the power of sin and is a present and continuous process of believers becoming Christlike, accomplished by the Holy Spirit’s power and presence. Sanctification represents a believer’s victory over the flesh, the world, and the devil:

Wretched man that I am! Who will deliver me from this body of death? Thanks be to God through Jesus Christ our Lord! So then, I myself serve the law of God with my mind, but with my flesh I serve the law of sin (Romans 7:24-25).

For everyone who has been born of God overcomes the world. And this is the victory that has overcome the world—our faith (1 John 5:4).

Submit yourselves therefore to God. Resist the devil, and he will flee from you (James 4:7).

Justification happens when God declares a guilty sinner to be righteous and is a once-time act. Sanctification, on the other hand, is a continuous process that releases the believer from the power of sin. By the power of the Holy Spirit the believer abides in Christ:

He saved us, not because of works done by us in righteousness, but according to his own mercy, by the washing of regeneration and renewal of the Holy Spirit (Titus 3:5)

To “abide” is to live, continue, or remain; so, to abide in Christ is to live in Him or remain in Him and is the position of all true believers. Jesus said:

I am the vine, you are the branches; he who abides in me and I in him, he bears much fruit, for apart from me you can do nothing. If anyone does not abide in me, he is thrown away as a branch and dries up; and they gather them, and cast them into the fire and they are burned. If you abide in me, and my words abide in you, ask whatever you wish, and it will be done for you (John 15:5-7).

The results of abiding in Christ include obedience to Christ’s commands (John 15:10; 1 John 3:24); following Jesus’ example (1 John 2:6); living free from habitual sin (1 John 3:6); and the awareness of a divine presence within one’s life (1 John 4:13).

The cause is Christ Jesus. Because he was obedient unto death, the way is open for believers to be forgiven of their sins and declared righteous before God. Justification delivers the believer from the penalty of sin, accomplished by Christ Jesus at the cross (Romans 5:18). As we abide in Christ, the ongoing process of sanctification transforms the life of the believer.

The word used in the scripture regarding the matter of the first humanity, the removal of the first humanity, the release from law, and the eradication of sin is 'reconciliation' (commonly so stated, though I prefer 'restoration' myself as it is not a matter of emotional 'reconciling' but a matter of restoration under Headship). The words in Greek are katallasso and apokatallasso.

The matter of sin is dealt with in Christ and in his death.

See Romans 5:12-21 for example. Sin itself is eradicated in Christ when he was 'made sin'. 2 Corinthians 5:21.

Thus made sin (or 'effected' sin, as it is poeio make/do) he died, taking sin itself down into death. Our old humanity is crucified with Christ. The removal of the sin of the world is a doctrinal matter, not a matter of Christians 'not sinning'.

'But now we are delivered from the law, that being dead wherein we were held ; that we should serve in newness of spirit, and not in the oldness of the letter', Romans 7:6.

God, sending his own Son in the likeness of sinful flesh and about (it is peri) sin, condemned sin in the flesh, Romans 8:3. Here, Paul describes the Son coming both in the likeness of sinful flesh (his incarnation) and 'about' sin, that is to say, being 'made' (or 'effected') sin - see 1 Corinthians 5:21 - sin itself was associated with him (though he were sinless) in the sight of God.

In death, that sin was taken into death, and was no more, It was eradicated. That is to say the 'sin of the world' for 'Behold the Lamb of God who taketh away the sin of the world', John 1:29. He takes it away, himself. (It is not taken away by the deeds of others).

This is what is revealed by the brasen serpent, John 3:14, in the wilderness. They did but look and they were healed from the plague, the biting of the serpent, Numbers 21:9.

All we have been bitten by the serpent. All have gone astray. Sin is within all of us.

Nor can we eradicate it by our own holiness. Paul discovered this as he tells us in Romans 7 - he had no power over sin as long as he was under law. For the law is the strength of sin, 1 Corinthians 15:56.

The harder one fights, the stronger it becomes. O wretched man that I am ! cries the sinner under law, Romans 7:24.

But God be thanked, there is a Deliverer - 'through Jesus Christ my Lord'.

For He is the One who takes away the sin of the world.

Behold the Lamb of God, is all that is required, just as they in the wilderness did but look at the serpent that was lifted up as an 'ensign'. So was the Son of man lifted up and so shall they that look upon him live, and be delivered from sin that dwells in flesh.