According to Wikipedia,
"A pure mate is a checkmating position in chess in which the mated king
and all vacant squares in its field are attacked only once, and
squares in the king's field occupied by friendly units are not also
attacked by the mating side (unless such a unit is necessarily pinned
to the king to avoid it interposing to block the check or capturing of
Essentially, a pure mate is a position in which each square around the opposing king is attacked only once by the mating side, so your first example is indeed a pure mate. However, your second example is not. While king can be blocked off by their own units, they cannot be attacked unless they are pinned, which they aren't.
To summarize, the Black king can be surrounded by its pawns and pieces so long as they are not attacked, and if they are, they must be a pinned. Furthermore, the pin must preventing the pinned piece from interfering with the mate.
As far as I can tell, multiple pieces may be on the board so long as the stipulations of a pure mate are met. The enemy pieces surrounding the king’s squares can be attacked, so long so as the mating piece is actually capable of mating, i.e. it cannot be captured or blocked.
Wikipedia gives this example from the final position from the famous Evergreen Game.
[Title "”The Evergreen Game,” Adolf Anderssen-Jean Dufresne. Berlin GER, 1852"]
[FEN "1r3kr1/pbpBBp1p/1b3P2/8/8/2P2q2/P4PPP/3R2K1 b - - 0 1"]
In this position, each of the Black king's available squares that it could move to are attacked exactly once by the White pieces, and the black blocking pieces are not attacked by white pieces at all. Although the White pawn that blocks off the g7 square for the black king is attacked, this is a real pure mate from high level play nonetheless.
If you wish to see a constructed position for a pure mate, @DM has made a great position for his answer to this similar CSE question.