Death in the Bible is always pictured as a separation between two things.
Physical death, spirit for body - Eccl 12:7 then shall the dust return to the earth as it was: and the spirit shall return unto God who gave it.
James 2:26 as the body without the spirit is dead, so faith without works is dead also.
Spiritual death, spirit from God - Isa 59:1-2 But your iniquities have separated you between and your God, and your sins have hid his face from you, that he will not hear.
Jesus died physically & spiritually on the cross:
Jesus died spiritually when He cried, "Eloi, Eloi, lama sabachthani? Which is, being interpreted, My God, my God, why hast thou forsaken me?
Watch out now, you start talking like that and there will be those that will ACCUSE you of NOT BEING SAVED. for what YOU have offered is that UPON the 'cross' God truly DID abandon Christ to DIE for our sins. That God was UNABLE to even LOOK upon His OWN SON once ALL the sin of the WORLD was placed upon His shoulders. And FOR THIS TO BE TRUE, then Christ could NOT have BEEN God Himself.
He was separated from God which is spiritual death.
Jesus died physically when He cried with a loud voice, and gave up the ghost. Mat 27:50, Luk 23:46, John 19:30.
The "sin nature is another subject that I would get into later, of which I will say, neither we or Jesus received a sin
nature.
However, Jesus did have as real flesh as you & I, that is sinful flesh.
Romans 8:3 For what the law could not do, in that it was weak through the flesh, God sending his own Son in the form of sinful flesh, and for sin, condemned sin in the flesh:
&
Philip 2:7-8 But made himself of no reputation, and took upon him the form of a servant, and was made in the likeness of men:
8: And being found in fashion as a man, he humbled himself, and became obedient unto death, even the death of the cross.
The way that we might have Eternal life is to know Jesus
John 17:3
We were reconciled to God by the death of his Son Jesus
Romans 5:10
Therefore it was both spiritual death and physical death.