I want to start by stating that the typical casual understanding of the "Christian" Afterlife is not based on The Bible at all but on Zoroastrianism, which influenced Manicheans like Augustine.
That some preachers continue to play ignorant of how Revelation even in English clearly depicts Hell and the Lake of Fire as distinct really annoys me, but there is no helping them.
The worth discussing issue is of Hades and Gehenna.
A total of three Greek words are translated "Hell" in the KJV translation of the New Testament. Tartaros is a word used only once in 2 Peter so not worth much of an in-depth discussion here.
Hades is the word that is clearly the synonym for the Hebrew Bible's Sheol (every OT usage of Hell is Sheol, but it also gets translated Grave and Pit, which also have other Hebrew words).
It's also the only of the three words used in Revelation. Paul also used the word in 1 Corinthians 15:55 once where the KJV renders it Grave rather then Hell.
So on a practical level what Hades refers to is the least disputable.
Etymology wise it is the most justifiable one to translate Hell as the origin of Hell is the equivalent concept from Germanic/Norse languages and mythologies. (Sheol, Hades and Hell were also used by the Pagans of their respective languages to refer to an Underworld abode of the Dead that has a god and/or goddess ruling over it that was sometimes called by that same name.)
But in terms of what your typical Preacher means by Hell in modern America, The Lake of Fire is clearly what they mean, and their understanding even of that concept is of disputable accuracy.
The third word is Gehenna. Gehenna is never by the KJV at least translated anything other then Hell. It is used 12 times, 11 of them are in the Synoptic Gospels and the only remaining one is in James Epistle.
I have for a long time and mainly still do view Gehenna as referring to the same thing as the Lake of Fire, but I'll get more into that later.
Many radical KJV onliers continue to insist Gehenna and Hades must be the same thing if the KJV used the same term for both. Though frankly I suspect the Latin translations of The Bible are the origin of both being translated the same way, the Vulgate doesn't use an expected Latin equivalent of Hades (like Pluto or Orcus) but rather Inferus, related to a word for fire.
Revelation 20 makes clear Hades is emptied and then thrown into the Lake of Fire before the New Heaven and New Earth are created and New Jerusalem descends.
The only passage using Gehenna that seems to be quoting or paraphrasing an Old Testament passage is Mark 9: 43-48 which is drawing on Isaiah 66:24. Isaiah 66 is still the same Prophecy as 65, which is clearly what Revelation 21 is drawing on, so this is contemporary with when the New Heaven and New Earth will be created. So it can't be describing Hades (Also Sheol is not used in the verse) but could fit the Lake of Fire.
Now many who have no trouble with Gehenna and Hades referring to different things may still desire to see the Lake of Fire as distinct from either. But the timing of Isaiah 66:24, as well the affiliation of Fire with Gehenna (at least one Gehenna passage also uses a Greek word for Fire, the verses in Matthew referring to the "Hell Fire"), as well as Gehenna being linked to the word "Damnation" all tell me Gehenna is probably the Lake of Fire.
Luke 16's parable of the Rich Man and Lazarus is the only passage linking Torment or Fire to Hell where Hades rather then Gehenna is used.
Deuteronomy 32:22 is the only passage linking fire to Sheol. KJV onliers will often call this the first reference to "Hell" in The Bible since Sheol's appearances in Genesis are all translated Grave in the KJV. But this verse does not attribute this Fire to tormenting the currently dead but does possibly connect it to future Judgment. The only difference between Sheol and NT Hades is Sheol might be a broader term that includes more of the depths of the Earth then just where the dead is/was. And thus may include Tartaros/The Abyss. And also could include the very center of The Earth, which modern Scientists believe is hotter then the surface of the Sun.
Ezekiel 32 describes Sheol, it describes the Dead being there, seems to imply they are awake there and thus contradicting the Soul Sleep view. But no references to torment or torture or burning.
So since Luke 16 is the only basis for Hades being a place of torture or burning torment with no second witness, you can't build doctrine on it.
The arguments against it being a Parable I find unconvincing. The main one is an assumption that Parables don't name names. But many view Job as a parable.
What is possible is that the Parable of Lazarus and the Rich Man was Jesus using the Pharisees own beliefs rhetorically against them, and not exactly meant to tell us the true nature of the After Life.
Gehenna is also the name of a real location outside Jerusalem. The place known in the Hebrew Scriptures as the valley of Hinnom, a place often linked to Child Sacrifice and burnt offerings to Molech. I've also seen it claimed that in Jesus time it was basically a garbage dump where garbage was burned.
Some think using that name for the place of Aionos punishment is merely a figurative or symbolic analogy. Others see this clear earthly location being linked as an argument against it being the Lake of Fire. I think it's possible where ever the Lake of Fire is cosmologically, it will at the White Throne Judgment be opened in the valley of Hinnom.
I think the possibility of Jeremiah 7:31-32 and 19:6 corresponding to Isaiah 66:24 would make a lot of sense.