For me, as a Christian, belief is a synonym for trust. ... For me belief is trusting that Jesus can actually make a difference in my life by advocating for me with God the Father so that I might be accepted by God, not in my own futile and failed attempts at righteousness. ... I believe (i.e. trust) that Jesus righteousness will be imputed to me and thus I will find favor with God the Father because of that relationship and for no other reason.
It seems to me what you are defining here is "Faith". See Hebrews 11:1
"Now faith is assurance of [things] hoped for, a conviction of things not seen." Romans 3:28
"We reckon therefore that a man is justified by faith apart from the works of the law.' and Galatians 2:16
"...yet knowing that a man is not justified by the works of the law but through faith in Jesus Christ, even we believed on Christ Jesus, that we might be justified by faith in Christ, and not by the works of the law: because by the works of the law shall no flesh be justified."
According to Webster's dictionary, faith is:
1. Confident belief in the truth, value or trustworthiness of a person, idea or thing.
2. Belief not based on logical proof or material evidence.
3. Loyalty to a person or thing.
Christians put their faith in Jesus and rely upon his righteousness and sacrifice on the cross for their salvation. However, Muslims believe that Jesus (pbuh) will disassociate from those who worship him instead of Allah.
Quran 5:116-118
After reminding him of these favors, Allah will say: "O Isa (Jesus) son of Maryam (Marry), Did you ever say to the people, "worship me and my mother as gods beside Allah?" He will answer: "Glory to You! How could I say what I had no right to say? If I had ever said so, you would have certainly known it. You know what is in my heart, but I know not what is in Yours; for You have full knowledge of all the unseen. I never said anything other than what You commanded me to say, that is to worship Allah, Who is my Rabb and your Rabb. I was a witness over them as long as I remained among them; but when You called me off, You were the Watcher over them and You are a Witness to everything. If You punish them, they surely are Your servants; and if You forgive them, You are Mighty, Wise."
In stark contrast Muslims put their faith directly in the Mercy of Allah for their forgiveness and salvation from the fire. We have the hope of forgiveness, but unlike Christians I don't have the assurance of salvation because Judgment Day may show that my testimony was hollow and that my life was hypocritical.
You may find it odd that, if you were to ask me, "If you were to die tonight, where would you go - heaven or hell?" or "Are you saved?" I would have to say, "I don't know, but I have the hope of salvation." The Quran repeatedly promises salvation for those who believe and do good works, but who on earth can judge his own faith as sincere and his good deeds as being done with the right intentions - much less anyone else? Only Allah knows where we will end up according to His Divine Preordainment, or Qadar - a fundamental Islamic Article of Faith.