Yes, the terminology on this topic means different things to different people.
And we can't go by what people think Biblical Concepts mean, we must seek to understand what the Bible means when it presents those concepts (eternal separation being a big one, lol).
From where do you draw the conclusion that all persons exist forever? This is the crux of the issue. Plato taught this, but I haven't seen this in the scripture.
From the teachings of Christ and the New TEstament Writers.
I'm out of time so for now I will just point out that when Christ taught about Hell (Gehenna) He teaches in terms of an eternal existence.
For example:
Matthew 25:46
King James Version (KJV)
46 And these shall go away into everlasting punishment: but the righteous into life eternal.
Both fates are identical.
The understanding of "death" in Scripture is something I don't think many put too much thought into. Those who erroneously teach annihilation view the Second Death as a cessation of existence, but the majority of teachings concerning eternal separation/eternal judgment do not imply that, but distinctly teach a state of existence, even though one is termed "dead."
Here is an example of a state of destruction without cessation of existence:
Matthew 10:6
King James Version (KJV)
6 But go rather to the lost sheep of the house of Israel.
(see also Matthew 15:24)
Now see that same word in relation to eternal judgment:
Matthew 10:28
King James Version (KJV)
28 And fear not them which kill the body, but are not able to kill the soul: but rather fear him which is able to destroy both soul and body in hell.
When Christ arrived Israel was in a state of destruction not terribly unlike the destruction awaiting those who die outside of God's grace. They had "life," but they did not have eternal life. Likewise, when the lost go into Hell, they will be dead, just as Israel was (the condition all men are born into)...but that does not mean they cease to exist.
On the surface those who teach annihilation seem to have a good case to present (Matthew 10:28 being their best proof-text), but, when you dig a little deeper you find that the "traditional" view that the lost will suffer torment for eternity is the correct position after all.
It's a lengthy study, and I think most would be surprised at just how much there is that has to be considered. Demons understand their fate:
Matthew 8:28-29
King James Version (KJV)
28 And when he was come to the other side into the country of the Gergesenes, there met him two possessed with devils, coming out of the tombs, exceeding fierce, so that no man might pass by that way.
29 And, behold, they cried out, saying, What have we to do with thee, Jesus, thou Son of God? art thou come hither to torment us before the time?
The torment they speak of gives no allusion to a cessation of existence, quite the opposite.
And the fate of demons and the lost is the same:
Revelation 14:9-11
King James Version (KJV)
9 And the third angel followed them, saying with a loud voice, If any man worship the beast and his image, and receive his mark in his forehead, or in his hand,
10 The same shall drink of the wine of the wrath of God, which is poured out without mixture into the cup of his indignation; and he shall be tormented with fire and brimstone in the presence of the holy angels, and in the presence of the Lamb:
11 And the smoke of their torment ascendeth up for ever and ever: and they have no rest day nor night, who worship the beast and his image, and whosoever receiveth the mark of his name.
Revelation 20:10
King James Version (KJV)
10 And the devil that deceived them was cast into the lake of fire and brimstone, where the beast and the false prophet are, and shall be tormented day and night for ever and ever.
So it is just my opinion that while we might not beat people over the head with Hell and eternal punishment, neither do we fail to be honest about what Scripture teaches. Eternal Judgment is one of the primary issues the Comforter has come to convict men about...
John 16:7-9
King James Version (KJV)
7 Nevertheless I tell you the truth; It is expedient for you that I go away: for if I go not away, the Comforter will not come unto you; but if I depart, I will send him unto you.
8 And when he is come, he will reprove the world of sin, and of righteousness, and of judgment:
9 Of sin, because they believe not on me;
...so if it is important enough for Christ to specify this issue, it isn't hard to believe He does so through us when He ministers through us. And note that it is the world He convicts. The convicting Ministry of the Comforter is clearly described as towards those who do not (yet) believe on Christ. This is the ministry the Comforter performs in this Age, so we would think that His message would be found in the message we convey in our efforts.
God bless.