The two "die" words are indeed the same, just used differently in the different parts of the text. Not only that, they refer to the exact same event, and speak of that event with the same meaning. The difference we see is in their usage.
For me, Romans 6 is my "desert island chapter" - it profoundly explains what takes place when a person is born again, and furthermore, it equally profoundly explains what it then means to live as a born again believer, a Christian.
Needless to say, at the heart of it is Jesus, with his own unique death and resurrection. He died, then he was raised to new life. His was the true "born again" - truly new life out of true death. Our "born again" experience is really a participation in his. That participation is real and bears fruit in our lives. It isn't as if it all happened in Jesus, and we end up simply as interested spectators in his experience. The new life he received is shared with us now. Paul goes on in chapter 6 to provide a stunning point to seal this logic:
12 THEREFORE, do not let sin exercise dominion in your mortal bodies, to make you obey their passions. 13 No longer present your members to sin as instruments of wickedness, but present yourselves to God as those who have been brought from death to life, and present your members to God as instruments of righteousness. 14 For sin will have no dominion over you, since you are not under law but under grace. (NRSV)
Our lives, declares the apostle, can now live for God, in obedience to him. Only because Jesus did it all, and we benefit radically because at the moment of becoming a Christian, we were bonded with Jesus in such a perfect way that when he died, our "old self" - a loyal slave of the unrighteous anti-God realm - died. That's part 1, if you like. If things stopped there, God's promised Kingdom simply could not happen. For God's Kingdom to come to life, with countless children of God praising him and performing his will, Jesus needed to be raised to new life. And that new life would be given to us - our "new self", a loyal child and citizen of the righteous kingdom of God.
Imagine a medical sci-fi scene as a metaphor: the Messiah on the operating table, still alive, with every one of his people joined to him. Veins, arteries, nerves, muscle, bones, everything dependent on his living body. Then, the Messiah dies. His system shuts down, and everyone bonded to him dies accordingly. On the third day, the miracle-working surgeon revives the Messiah. In the process, he gave the Messiah a whole new body - an eternally living body - and now the Messiah can be the risen lord over all things forever. And with that, we also needed to be given new bodies, that could reign with Jesus. And as new life pulsed into the Messiah, it also pulses into us. Because:
IF we have died with Christ, we believe that we will also live with him.
Romans 6:8 introduces an "IF ..., THEN ..." logic, a cause & effect statement. Verses 6-7 made the point that our death - our participation in Jesus' death - was final and absolute, to the point that our old self was dead, and our slavery in the unrighteous anti-God realm has finished. Now, IF we died so absolutely by participating in Jesus' death, THEN we will absolutely share in his new life as well! It is certain logic. The perfect cause & effect. So "die" in each verse refers to the exact same point, our dying with Jesus because we had been bonded with him.
Now, may I please make a point about being baptised into Christ's death, from 6:3?
Firstly, consider carefully the logic of all these verses. Conclusion: if you want to participate in Jesus' death and resurrection, and therefore be saved, you must be baptised into Jesus. There's no getting around that logic. Therefore, if someone is not baptised, that someone is not, cannot be saved. Is this really the case???
I thank Martyn Lloyd-Jones for resolving this logical - and theological - dilemma for me. Romans 6:3 does not, cannot, be referring to the water baptism exercised after someone becomes a Christian. Baptism does not save us - it symbolises and celebrates the salvation we have already received by having already participated in Jesus' death and resurrection at the very point of faith. So what does 6:3 refer to? The baptism of the Holy Spirit whom we receive when we become Christians. The Holy Spirit bonds us into Jesus, making that "spiritual" union with Jesus, so that we then participate in his death and new life.
I hope this helps, and I (sort of) apologise for the length.