There are basically two contrasting faiths in regards to Abraham sacrificing his son. The two views are between the Christian faith and then the similar Jewish and Muslim faiths. The second contrast is over which son was being sacrificed. We will look at the second contrast first.
Some of the Muslim faithful believe Abraham was sacrificing Ishmael, while others believe it was Isaac, as in the Christian and Jewish faiths. The source of the two different beliefs is that Ishmael was the first born, the righteous son. But for Christians, the first born of the promise was Isaac. So, it was Isaac being sacrificed. In turn, this motif plays out in Christianity in a number of ways. For example, the role of the promise and faith therein for one’s salvation is indicated.
The primary contrast in all of this, however, is the view of the sacrifice itself. For Jewish and Muslim faiths, it appears they believe the sacrifice would not happen in the end. For Christians, the sacrifice would happen.
To rephrase the contrast, was Abraham actually sacrificing his son (leaving alone the question of which one)? For the Jewish and Muslim faiths, the answer is a qualified yes. Yes, it was happening, but Abraham would not go through with it. G-d would intervene. It was only a very serious test.
The swoon theory that views Christ's sacrifice on the cross is one result of that belief structure. This theory is that the sacrifice of Christ, like with Abraham's son, only appeared to happen, but Jesus was revived in the cave, never died on the cross, and never resurrected out from the dead on the third day. They may believe he was a good man or a prophet of G-d, but not Messiah.
For the Christian faith, the answer is absolutely yes. The son was as good as dead as they traveled up the mountain to the altar. The son is on the altar, Abraham’s arm raised, and but for G-d, it was over.
By faith Abraham, when he was tried, offered up Isaac: and he that had received the promises offered up his only begotten son, Of whom it was said, That in Isaac shall thy seed be called: Accounting that God was able to raise him up, even from the dead; from whence also he received him in a figure.
So for Christians, the sacrifice of the promised seed in Isaac was a real sacrifice. To be sure, it did not happen with Isaac, but it was a clear foreshadowing of the reality in Messiah to come who would die and be resurrected out from the dead. Even if G-d had not stayed Abraham’s hand, G-d would have raised Isaac back from the dead. That sequence of course then took place in Jesus Christ who was sacrificed as the promised righteous seed. G-d raised Him from the dead to everlasting life, as He will for all those who believe.