At the fall, Genesis 3, all three aspects of death came with it. Before the fall, Adam was not subject to any form of death, but through his sin, death became a grim reality for the entire human race. The death Paul refers to in Romans 6:23 are spiritual death as well as eternal death, but not physical death. The wages of sin is separation from God both in the temporal and the eternal.
1 And you were dead in your trespasses and sins, 2 in which you formerly walked according to the course of this world, according to the prince of the power of the air, of the spirit that is now working in the sons of disobedience. 3 Among them we too all formerly lived in the lusts of our flesh, indulging the desires of the flesh and of the mind, and were by nature children of wrath, even as the rest. Ephesians 2:1-3 (NASB)
The fallen nature of man does not understand death. Depending on culture, some fight to stay alive with their all. The correct understanding of death comes to us from God’s Word. The word death is used in three different ways in Sacred Scripture. The first is spiritual death, which is separation from God (Ephesians 2:1,1). Then, of course, there is physical death (Hebrews 9:27). Then there is eternal death, which is referred to as “the second death” in Revelation 20:11-15). While many would consider the topic of death to be a subject to be avoided, it is vital for us to have a clear understanding of all three manifestations in biblical terminology so that we will have a better foundation for sharing the Good News.
23 For the wages of sin is death, but the free gift of God is eternal life in Christ Jesus our Lord. Romans 6:23 (NASB)