death metally melodic death metal and "the sort of melodic death metal that is death metal" would be more old school stuff like At The Gates, Like An Everflowing Stream, and TJR and Whoracle. (Lunar Strain and Subterranean are honestly closer to melodic black metal than death metal).
"Melodic metal" means stuff like Firewind. If someone says melodic extreme metal, they mean stuff like Children of Bodom, Wintersun, and Kalmah, although that stuff is also sometimes called melodeath (distinct from melodic death metal).
"not the true death metal melodic death metal, but the sort of offshoot of melodic death metal" - this refers to stuff that isn't old school death metal melodic death metal as defined in the first example, but is also not melodic extreme metal. It's an evolved form, currently played by Soilwork, Scar Symmetry, In Flames, Dark Tranquility, and others. I call it melodic alternative metal.
Alternative metal is metal that incorporates some elements of hard rock and possibly nu metal. A very varied sound, because it's kind of a category for stuff that doesn't fit. If it's extreme metal (uses blast-beats, really heavy guitar work, almost entirely extreme vocals, etc) it's NOT alternative metal. There's overlap between what's just plain Heavy Metal and Alternative Metal.
