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
So essentially, a pure mate is a position in which each square around the opposing king is attacked only once by the mating side, as you said, so you are correct with your first example.
However, your second example is not a pure mate. While pieces of the color of the opposing king are allowed to surround it, they cannot be attacked unless it is a pin, and none of them are pinned. The white knight on h5 and the white rook on e7 both attack a black pawn that is on a square adjacent to the black king, and the pawn is not pinned. Under the stipulations of what a pure mate is, your second position is thus not a pure mate.
So basically, 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.
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 of the famous Evergreen Game.
This is the final position of the 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 b7 square for the black king is attacked, this is a real pure mate from professional 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.
I hope that this helps you!