Since the Second Death is a Bible FACT, and those who do not believe on Christ are subject to wrath (John 3:36) it follows that when Christ bore the wrath of God on the cross, and was temporarily forsaken of the Father, that was indeed the suffering of the Second Death. The Second Death is eternal SEPARATION from God and Christ in the Lake of Fire. And Christ suffered for our sins, not just physically, but in body, soul, and spirit.
But, to your point about the scriptures, yes, you are correct that the author's understanding is that Christ paid a price, suffered an unjust penalty, was a substitute, etc.. But, these are allusions to what the scapegoat was, or stood for, in the Levitical atonement ritual. The Israelite's sins were transferred to the animal, both those that were killed and their blood was sprinkled for cleansing, and to the one left to wander and disappear in the desert. Jesus fulfills these allusions in all that the he did on the cross, but appreciate the fact that all these practices were figurative, obviously! The blood of bulls and goats have nothing to do with the sins of man, but God allowed them to be, due to the figurative sense that they conveyed. God forgave the Israelites provided that the practice was fulfilled, not because someone or something suffered a penalty commensurate to eternal death. For fundamentally, this is impossible, obviously!
But again, by ending the Law or Old Covenant, by death, all the definitions and descriptions of what the inspired writers wrote about Christ, are true, in and of themselves, contextually speaking. But since no one can suffer eternally, and at the same time be in the Kingdom, then obviously one has to interpret these verses in their specific context.
N6, you're being frivolous in your interpretation of scripture, just because Jesus quoted Ps 22:1, it does not mean that he suffered eternally, as you are eisegetically trying to claim. For example, many scholars do not believe that he was forsaken, but that his statement was alluding to the periscope in its entirety, but that he referenced it by its first line (common Hebraic practice). Or, other scholars claim that the Aramaic properly interpreted is 'for this you have spared me'. Either way, even if taken literally, it says that he felt abandoned in his moment of agony, not that he was ostracized eternally.
In that, not one scripture that you quoted claims that Christ paid an eternal punishment, not one N6! All the descriptions of his physical death can be said of countless others, who died in a much more protracted and excruciating manner. Like the Christian martyrs, like the horrible tortures and death we read throughout history, even like the two malefactors crucified beside him. And the only remarkable thing about Christ's death compared to theirs, is that he didn't deserve it. This is where all the scripture that you quoted is derived from.