In my gameplay if you Long Rest on the surface then yes, you teleport to same beach/island.
But Long Resting in dungeon's changes the graphics for, well, an enclosed area. Withers and Goblin Slayer with Scottish accent if you activated them teleport to it

Yes, what I meant is that I've assumed that in each area we visit, we identify a safe spot to rest, and whenever we rest in that area we return to that place, as opposed to the same graphic standing in for different notional locations around the map. So it's the main camp up in the overland principal EA area, a different spot in the Underdark, and so on. And I'm hoping that when we move on to another major area then we'll get a different camp graphic. And that there might be separate "camp" graphics too for inns that we choose to spend the night at later.
Of course, my headcanon may not work perfectly as I suspect that various small underground areas possibly share the same "minicamp", though I've never paid a whole lot of attention and tend not to rest outside the larger areas anyway

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