>I don't really think the name matters; Jesus wasn't speaking English.
Diction mattered a great deal to Christ. Gehanna (Hebrew: Valley of Hinnom; גיא בן הינום) was a location in Christ's time, that was likely associated with burning, destruction and loss - things that one might be expected to feel in the absence of the Creator.
Hades was where all dead went and remained until the day of judgment. Christ's reference to it as Paradise implies it isn't a place of suffering. Catholicism's Limbo implies it is a place of waiting.
The modern notion of Hell as a location dedicated to the eternal suffering of man, is quite different from either of those places.
Jesus said "Then they will go away to eternal punishment". That seems clear enough to me that there's some place (not necessarily a physical place) where people will suffer eternally. I think that covers the basics of the Christian view of Hell. Whether it's associated with burning or not seems a lesser matter to me.
This doesn't seem relevant to the original question of whether the Bible says Hell exists or not. People can agree that something exists without agreeing how it was created.
One view[1][2] is that Hell isn't really a place, but rather a state of being, and the primary suffering of Hell is the separation from God[3]. God didn't create it, rather we ourselves created it by separating ourselves from God through sin.
> Then they will go away to eternal punishment, but the righteous to eternal life.