I believe Saro is correct, but want to provide some additional comments. A lot of the answers are launching into a theological interpretation of Adam's death, which is fine, but I'd like to just focus on understanding the text of these passages by providing some background material, so we can understand better what these passages are saying, regardless of our overall theology taken from many different passages.
In Hebrew the doubling is used for emphasis. So when you see something like "gold gold", you might translate this as "pure gold" or "fine gold" to emphasize how extremely goldy it is. Of course translators then struggle in translating this idiom, which is why in translating "die die" you see "surely die", "certainly", "in dying you shall die" (which I believe is a poor translation as it elaborates a bit too much).
This doubling occurs over a hundred times in the old testament, and when it does
most often translators try to understand what is being emphasized and then express that, as in the following:
Gen 14.10: Now the Valley of Siddim [tar tar] pits. translated as "was full of tar pits"
Gen 18.10: And they piled them in [heaps heaps], and the land stank. translated as "countless heaps"
Numbers 14.7: “The land that we went through to explore is [very very] good. translated as "exceptionally good"
Eccl 7.24: Whatever is—it is far [deep deep]. Who can discover it? translated as "beyond comprehension"
2 Chron 31.6 And they gave [heaps heaps]. translated as "heaps upon heaps"
At other times, the doubling is included verbatim as the english doubling also conveys an emphasis.
Joel 4.14 Commotion, commotion in the valley of decision! For the day of Yahweh is near in the valley of decision!
Prov 20.14 “Bad, bad,” the buyer will say, but when one goes to him, then he will boast.
Eccl. 1.2 “Vanity of vanities!” says the Teacher, “Vanity of vanities! All is vanity!”
2 Chron 23.13 And Athaliah tore her garments and cried, “Conspiracy! Conspiracy!”
Sometimes the doubling is translated as "every":
- 2 Chron 13.11 They offer burnt offerings to Yahweh [morning morning] and [evening evening] translated as "every morning" and "every evening".
And finally sometimes the translators don't know what is being emphasized so they ignore the doubling:
Lam 1.16: For these things, I am weeping, my [eyes eyes] flow with tears translated as a single "eyes".
So I believe it is a mistake to read a certain special kind of death into these passages, but rather to point out that death was being emphasized in the passage. If they had the technology, then this could have been written in bold font, or underlined, and then modern texts could bold that word and it would be a faithful translation. That would then still leave open the theological implications, but those would need to be determined from other passages.
The Divine Attention
When looking at prophecy or statements of the divine (which both of the above are), I think it's helpful to keep in mind that these are handled differently than a modern reader would expect. We have a fairly flat, almost geometric view of God -- omniscient, omnipresent, etc. The cost of that is that if God knows everything, it's hard for us to imagine him knowing some things more than others. Our view of God is crippled by the infinities we ascribe to him. So we have developed a theology around that which doesn't give much room for the narrative descriptions of God in the Bible, for example God asking Adam where he is, or saying that he knows Abraham (wouldn't he know everyone?), etc. God as portrayed in the Bible is less passive. He looks at some things and hides his face from others. He knows some people and not others. He pays attention to some things but not others. He goes down to earth to see what the people at Babel are up to. Note that I am not making a christological or theological argument describing God's abilities, but describing how God is portrayed in the text, which is what we need to understand.
One such characteristic portrayal is of the divine view. As opposed to man, God sees the end of things, whereas man sees the process. So someone about to die can be described as dead. Someone about to live can be described as alive. People also have this idiom "dead man walking", "you're dead" (said to a live person). We use this idiom when there is no doubt in our mind as to the person's fate. But God sees ends much more clearly than we do, so these expressions can be used even if the death wont occur for a long time as recockend by man. In God's view, the moment the man's end shifted from life to death, that's the moment he became a dead man, and so the man died. And when a man repents, that is the moment his end shifted out of the realm of death, and so he lived.
I believe this is how to read these passages, regardless of the theological interpretations of the types of death or what those deaths signified. The passages are describing the shift from life to death and vice versa, and what creates those shifts. They are not elaborating on what that death means. For an understand of that, you need to look at other verses.
We also see this in the New Testament, where Jesus said Go away, because the girl is not dead, but sleeping. Matt 9.24 - when he was describing a woman that was dead but would be alive. The moment he decided or knew (the father decided) that she would be raised, that's the moment she was classified as being alive.
We also have Matt 22.32 “I am the God of Abraham and the God of Isaac and the God of Jacob”? He is not the God of the dead, but of the living!” Yet we know that these were dead, waiting to be raised after Christ was raised as the first-fruit. But that temporary death was not their end, the end of the patriarchs was life, and so God considered them alive.