I can tell you what it means to me as a Christian and a believer in Jesus Christ. Not sure about folks who have defined Higher Power differently. But for me and as a Christian it boils down to this. NOT telling anyone else what to believe or who to believe in. Not telling anyone what or who their higher power is or should be. This is mine and I'm just trying my best to answer the question clearly from my point of view.

Jesus paid the price for my sin in full. I am a new creature in Christ. The new person He has made me and His sacrifice for sin purchased for me, that new person is forgiven, is clean. And since Christ paid the price in full, that new person is secure in Christ. We are in the "in between time" if you will and I am still a product of a fallen world and my "flesh" is a product of that fallen world as well. But none-the-less it is true I am redeemed and "clean" because of the righteousness of another. That other is Jesus Christ. Here are a few of the Bible verses that explain what I am talking about.


Rom 5:8-18
8 But God demonstrates his own love for us in this: While we were still sinners, Christ died for us.
9 Since we have now been justified by his blood, how much more shall we be saved from God's wrath through him!
10 For if, when we were God's enemies, we were reconciled to him through the death of his Son, how much more, having been reconciled, shall we be saved through his life!
11 Not only is this so, but we also rejoice in God through our Lord Jesus Christ, through whom we have now received reconciliation.
12 Therefore, just as sin entered the world through one man, and death through sin, and in this way death came to all men, because all sinned--
13 for before the law was given, sin was in the world. But sin is not taken into account when there is no law.
14 Nevertheless, death reigned from the time of Adam to the time of Moses, even over those who did not sin by breaking a command, as did Adam, who was a pattern of the one to come.
15 But the gift is not like the trespass. For if the many died by the trespass of the one man, how much more did God's grace and the gift that came by the grace of the one man, Jesus Christ, overflow to the many!
16 Again, the gift of God is not like the result of the one man's sin: The judgment followed one sin and brought condemnation, but the gift followed many trespasses and brought justification.
17 For if, by the trespass of the one man, death reigned through that one man, how much more will those who receive God's abundant provision of grace and of the gift of righteousness reign in life through the one man, Jesus Christ.
18 Consequently, just as the result of one trespass was condemnation for all men, so also the result of one act of righteousness was justification that brings life for all men.
(NIV)


Rom 8:1-4
1 Therefore, there is now no condemnation for those who are in Christ Jesus,
2 because through Christ Jesus the law of the Spirit of life set me free from the law of sin and death.
3 For what the law was powerless to do in that it was weakened by the sinful nature, God did by sending his own Son in the likeness of sinful man to be a sin offering. And so he condemned sin in sinful man,
4 in order that the righteous requirements of the law might be fully met in us, who do not live according to the sinful nature but according to the Spirit.
(NIV)


Col 1:13-22
13 For he has rescued us from the dominion of darkness and brought us into the kingdom of the Son he loves,
14 in whom we have redemption, the forgiveness of sins.
15 He is the image of the invisible God, the firstborn over all creation.
16 For by him all things were created: things in heaven and on earth, visible and invisible, whether thrones or powers or rulers or authorities; all things were created by him and for him.
17 He is before all things, and in him all things hold together.
18 And he is the head of the body, the church; he is the beginning and the firstborn from among the dead, so that in everything he might have the supremacy.
19 For God was pleased to have all his fullness dwell in him,
20 and through him to reconcile to himself all things, whether things on earth or things in heaven, by making peace through his blood, shed on the cross.
21 Once you were alienated from God and were enemies in your minds because of your evil behavior.
22 But now he has reconciled you by Christ's physical body through death to present you holy in his sight, without blemish and free from accusation--
(NIV)