Metal fans, especially newer ones who were not there for the golden ages love to break metal down into finite genres and groupings, which leads to tremendous arguments about what a band "is" or "isn't". But the nature of metal before the crystallization of Death and Black metal between 88 and 91 or so shows that it is a flawed outlook. If you read old zines from the 80s with interviews from Slayer, Kreator, Possessed, Bathory, Sodom, Sarcofago, Sepultura, Venom and Exodus you will find quite random applications of "thrash" "death" "black" and "speed" metal between all of those bands, with many bands being called or claiming multiple titles in the same interviews. Exodus being called Black Metal. Bathory Death Metal etc. As best as I can tell, "death" became associated more with modern musical notions of Death Metal starting in 88 with Florida bands. Simultaneously, the Grindcore movement which later basically consciously converged with death metal started at the same time. Black metal was still being used often by many of these bands, and did not become associated with the modern notions until 1991 or so when Norwegians specifically differentiated it as a distinct genre.
So terms like "first wave Death Metal" and "first wave Black Metal" are interchangeable, before 88 there were absolutely no finite genres. Extreme metal was all one genre. How power metal fits into this construct I don't know exactly. Somewhere in the early 90s it became identified with less aggressive, more heavy metal oriented speed metal forms. I'd have to look at 80s power/speed metal interviews to get a better idea. But in the 80s it was also associated with the pre-genre super construct as exemplified by Metallica using the term.
Bands can be many genres at multiple times. Black Sabbath is hard rock, classic rock, psychedelic rock, heavy metal, and doom metal equally and at the same time. All genres are equally valid.
Heated genre arguments that try and place bands firmly into different baskets or support the post 2000 internet notion of a linear development of heavy>speed>thrash>death>black are short sighted and for newbies and/or posers. Metal and music does not evolve like species do, where there are no gene transfers between lineages, it evolves more like religion, yet even more amorphously.