Did the Second Person of the Trinity Die on the Cross or Just Jesus' Humanity?

I hear more often than not from knowledgeable Christians that although Jesus is God, God did not die on the cross – Jesus’ humanity did.

When bodies die they remain in the grave until resurrection but the soul remains conscious in the intermediate state doing what souls can do without a body. With that premise in view, how does the death of the Second Person of the Trinity impinge upon his divinity, authority, abilities or whatever? Was the death of the body sufficient to do away with Jesus’ sovereign rule over the universe? Was death even sufficient to stop the Rich Man (from Luke 16) from trying to correct God? One would have to ask how the Lord managed prior to the incarnation (when without a body) if we may not say that the Second Person of the Trinity truly died upon the cross.

In death Jesus was separated from his body but he was still conscious and active according to what he could do without a body. It is thought that if the incarnate Son died, then He couldn’t function as God, which is why I like to tease out that we believe in a conscious death, one that permits the person to operate on some level without a body. Christians are not annihilationists after all. In the case of the Son, Jesus operated most of his divine life without a body yet while ruling the universe. Accordingly, why not say he died and in that sense operated as before? I think some unwittingly impose an annihilation understanding of death onto their own thinking about the Savior's death, thereby not allowing the Savior-person to have truly died yet function as God while dead.

We must allow Scripture to inform us of what is possible. Christians should agree that if a divine person died on the cross, then death must be compatible with that divine person. The only question is why is the possession of the divine nature incompatible with the death of a person who possesses that nature? In other words, what would death of a divine person, who had assumed a human body and reasonable soul, prevent Him from doing or being? Obviously it would prevent him from doing physical things like walking but what else with respect to divine ability and ontology? In laboring the point we might note that death of a human person does not eliminate the human will and other things human, though it eliminates some things. Given that God does not require a physical body to function as God, why can’t the death of a divine person impinge upon the assumed physical properties of the person and not spiritual ones?

In the like manner, I’m willing to look at birth in such a way as to be compatible with a divine person coming forth from the Virgin's womb. Did Mary give birth to a divine person, or just a human nature? If birth implies the origin of someone new (a new person), then only humanity came forth in the virgin birth since the person born of the virgin always existed. However, Mary carried a person (and not just an embodied nature) in her womb, and after her water broke, she then labored to bring forth the person she had carried. In common parlance we call that giving birth. Since a divine person came forth, we must let that reality inform our understanding / definition of birth (rather then let our understanding of birth redefine what occurred in that manger in Bethlehem). Inception, let alone birth, need not precede the origin of a new person, precisely because the eternal Son of God, a person, was born of a virgin. When we start with Scripture apparent problems often go away. Question 37 of the Westminster Shorter Catechism puts it this way, or rather it simply assumes the point when making another:

“How did Christ, being the Son of God, become man? Answer: Christ the Son of God became man, by taking to himself a true body, and a reasonable soul, being conceived by the power of the Holy Ghost in the womb of the Virgin Mary, of her substance, and born of her, yet without sin.” (emphasis mine)

It's a comfort to know that the Second Person of the Trinity died for sinful people. If only Jesus human nature died, then only your nature would be saved, which stops short of the salvation of individual and distinct persons since the one nature is shared by all.