In what sense was Jesus's death a sacrifice?

God sacrificed his only begotten Son, Jesus Christ, to satisfy the debt we owe: Romans 6;23 For the wages of sin is death; but the gift of God is eternal life. If you choose not to accept his sacrifice, then you will pay the debt you owe with your life. However, your death will not be a sacrifice, it will be a needless loss.

2 Peter 2;9

The LORD is not slack concerning his promise, as some men count slackness; but with longsuffering to us-ward , not willing that any should perish, but that all should come to repentance.

He paid our sins by showing us that everything He has told us is, in fact, true and corect. That He can give up His life on the cross just to tell us that everything He taught us is right. If it is difficult for someone to surrender his life for something he claims as true, what more if not.

He, on the day of His crucifixion, could have taken back His Words or could have done miracles to save His life beforehand but opted not to because of His reverence for the Lord and His Will to save more of the humanity.

Latter-Day Saints (Mormons, or members of the Church of Jesus Christ of Latter-Day Saints), believe that Christ suffered not only on the cross but also in the Garden of Gethsemane.

Consider Luke 22:44:

And being in an agony he prayed more earnestly: and his sweat was as it were great drops of blood falling down to the ground.

A more detailed description of this suffering is given in the Doctrine and Covenants, in a revelation to Joseph Smith. Christ is speaking here (D&C 19:16-19):

16 For behold, I, God, have suffered these things for all, that they might not suffer if they would repent;

17 But if they would not repent they must suffer even as I;

18 Which suffering caused myself, even God, the greatest of all, to tremble because of pain, and to bleed at every pore, and to suffer both body and spirit—and would that I might not drink the bitter cup, and shrink—

19 Nevertheless, glory be to the Father, and I partook and finished my preparations unto the children of men.

The prophet Isaiah gave a description of the suffering of Jesus, who was yet to come, in Isaiah 53:

Verse 5 indicates he was wounded, bruised and received stripes, which I believe makes reference to when he was brutally scourged.

5 But he was wounded for our transgressions, he was bruised for our iniquities: the chastis*m*nt of our peace was upon him; and with his stripes we are healed.

Verse 10: his soul made an "offering for sin":

10 Yet it pleased the Lord to bruise him; he hath put him to grief: when thou shalt make his soul an offering for sin, he shall see his seed, he shall prolong his days, and the pleasure of the Lord shall prosper in his hand.

Verse 12: the phrase "poured out his soul unto death" sounds as terrible a sacrifice as any I can imagine.

12 Therefore will I divide him a portion with the great, and he shall divide the spoil with the strong; because he hath poured out his soul unto death: and he was numbered with the transgressors; and he bare the sin of many, and made intercession for the transgressors.

Please also consider the description in "The Purifying Power of Gethsemane". Here's an excerpt:

We do not know, we cannot tell, no mortal mind can conceive the full import of what Christ did in Gethsemane.

We know he sweat great gouts of blood from every pore as he drained the dregs of that bitter cup his Father had given him.

We know he suffered, both body and spirit, more than it is possible for man to suffer, except it be unto death.

We know that in some way, incomprehensible to us, his suffering satisfied the demands of justice, ransomed penitent souls from the pains and penalties of sin, and made mercy available to those who believe in his holy name.

We know that he lay prostrate upon the ground as the pains and agonies of an infinite burden caused him to tremble and would that he might not drink the bitter cup.

As near as we can judge, these infinite agonies—this suffering beyond compare—continued for some three or four hours.

You are right that he did receive the greatest possible reward, but only after the greatest possible suffering.

A final scripture explains that it was an infinite and eternal sacrifice, which we cannot fully comprehend, except by considering the guilt and anguish you and I feel when we sin, and multiplying that by an infinite factor: Alma 34:10,12,14.

In Jesus' prayer at Gethsemane (Mt 26:39), we see that Jesus was not happy with his cross:

Mt 26:39 (NIV)
Going a little farther, he fell with his face to the ground and prayed, “My Father, if it is possible, may this cup be taken from me. Yet not as I will, but as you will.”

The fact that He was "sorrowful and troubled" (Mt 26:37) and said that his soul was "overwhelmed with sorrow to the point of death" (Mt 26:38), shows that his death was a sacrifice.

You said:

To me, sacrifice would mean to act in the best interest of others when you know doing so could hurt you.

Being crucified is so painful that the Romans even offered pain killer before the cross. Jesus's death was a sacrifice because the pain of the cross was extreme, even if it was only for a limited time.

The basic story of the gospel is this: we, being sons and daughters of Adam, are sinners and because of our sin we deserve punishment from God because he is a just God. At the same time, Good loves us enough that he sent his own Son to both live the perfect life we could never live and to bare the punishment of our son, becoming the ultimate sacrifice. When he died, he didn't go to heaven but the opposite : he was separated from the Father as punishment for our sin. Because of his sacrifice, all you have to do to be saved from eternal damnation is to accept his offer to be your king, father and Lord.

Our at least that's my interpretation of the Protestant / Reformed gospel of salvation.

A sacrifice could mean to get something bad you could avoid, but you choose to get for a good reason. Now, Jesus sacrifice is well described in Philippians 2:5-8

5 Let this mind be in you which was also in Christ Jesus, 6 who, being in the form of God, did not consider it robbery to be equal with God, 7 but made Himself of no reputation, taking the form of a bondservant, and coming in the likeness of men. 8 And being found in appearance as a man, He humbled Himself and became obedient to the point of death, even the death of the cross.

Now, this Jesus did not deserve. In Roman 6:23 this is even clearer:

For the wages of sin is death, but the gift of God is eternal life in Christ Jesus our Lord.

Now the wages of our sins is death. Jesus never sinned (according to Hebrews 4:14-15), so He didn't deserve death. But we did! Yet He got our wages (death), while we receive His gift (eternal life).

The Bible gives us other informations regarding the consequence of sin, in Isaiah 59:1-2:

1 Behold, the Lord’s hand is not shortened, That it cannot save; Nor His ear heavy, That it cannot hear. 2 But your iniquities have separated you from your God; And your sins have hidden His face from you, So that He will not hear.

So, according to this text, our sins, our iniquities, create a separation between us and God. Now, when Jesus was on the cross, He said (Matthew 27:46, Mark 15:34):

"My God, My God, why have you forsaken me?"

Jesus was experiencing this separation, that He did not deserve, in our place, since "all have sinned and fall short of the glory of God" (Romans 3:23), so we all deserve being abandoned by God, but Jesus was abandoned and we are not. Now Jesus decided to do this, which was a real sacrifice for Him, because He wanted to save us from being separated from God and die.

He then rose from the dead and went in Heaven because death could not keep him:

23 Him, being delivered by the determined purpose and foreknowledge of God, you have taken by lawless hands, have crucified, and put to death; 24 whom God raised up, having loosed the pains of death, because it was not possible that He should be held by it. (Acts 2:23-24)

Death could not keep Him, since He was sinless (there are various passages that explain that the consequence of sin is death; e.g. Romans 5:12; Romans 6:23; Romans 7:5; 1 Corinthians 15:56; James 1:15). So Jesus, not deserving this, made a sacrifice dying a death He did not deserve and experiencing a separation from God He did not deserve. And He did so, so that we may have the opportunity not to experience such a separation from God and such a shameful death, as He experienced, because of our sins.

I suspect that most would consider it a "sacrifice" if I were to push someone out of the way of a train and in the process get hit by the train and die.

Using the logic in this question (as I interpret it), it isn't a sacrifice since I only sacrificed my mortal life and ended up in heaven with an eternal life because I'm a Christian.

If saving the mortal life of one at the cost of your own is a sacrifice, how much more would saving the eternal lives of all who will accept be a sacrifice?

...not to mention that I'd much rather get hit by a train that suffer the way Jesus did.

Not everyone believes that Jesus went to heaven when he died. In fact, there is a biblical basis and tradition which teaches that Jesus actually spent time in hell receiving the punishment for our sins before his resurrection. In that sense he made a sacrifice, but he was also a sacrifice in the sense that his blood atoned for our sins and he acted as a scapegoat for our sins.

The sacrifice of Jesus was many things.

It was a personal atonement for our sins, bringing salvation. But it is much, much more than only that.

It was a propitiation. God's wrath was/is hot upon the human race, his sacrifice was a shedding of innocent blood to put away that wrath. It was for us, but it was directed toward God. God required innocent blood, Jesus paid the debt...which allowed God legal right (his own standard) to forgive any who would turn to him.

In addition, the term sacrifice, points back to all the OT stuff which required a sacrifice to put away sin. He was the Lamb of God which took on the sin of the world. Notice, he is God's Lamb, not ours. God set his Lamb aside for death, he was accursed (set aside for judgment without hope of redemption) and sacrificed (blood shed for God to see...he was "put forth") in order for God to make his own way to interact with his sinful creation.

Also, the blood which was shed, not only allowed salvation to occur, but it is the same blood that God continues to view us through. Jesus is our high priest interceding for us even now and pointing to the blood, "your just requirements have been fulfilled...I paid if for them."

The sacrifice was indeed a sacrifice however. Not only in the pain and suffering of it all, but by virtue of the fact that he was GOD in the flesh...Immanuel. He had been surrounded and bathed in a pool of glory from eternity past, but in his coming here to die in our place, he left all of that, and became a high priest so he could understand what we go through (he became a perfect high priest). But the coming alone was a sacrifice. To be in a body of sinful flesh was a sacrifice. To live a life among sinners was a sacrifice. To be surrounded by people who didn't even care about him was a sacrifice. For the creator of heaven and earth, Jehovah God to be lonely, misunderstood, called a bastard, mocked, spit on, argued with, rejected, and crucified was truly a sacrifice. He was tempted in all points, just like we are. He sacrificed more than we will ever know. One drop of his blood was worth more than all of our blood put together. It was perfect blood, sinless blood.

To reduce the sacrifice to hell or heaven is reducing his will to nothing. True he did not end up in hell for eternity, but his entire existence on this planet was a sacrifice and it was his will to suffer for us.