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’s kings 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.
Now, if you want an example involving pins, the Immortal Pin Game comes to my mind. Take note of the final position.
EDIT: It turns out that, thanks to Brilliand, the game below does not end with a pure mate because Black’s knight is attacked twice. Whoops!
[Title "”On Pins And Needles,” Edinburgh Club Championship, 1981,Geoff Chandler-Richard Kynoch, 1-0"]
1. e4 c6 2. d4 d5 3. e5 Bf5 4. Nc3 e6 5. g4 Bg6 6. Nge2 c5 7. Be3 Nc6 8. dxc5 Nxe5 9. Nd4 Rc8 10. Bb5+ Nc6 11. Qe2 Nf612. O-O-O Be7 13. h4 h5 14. f3 Nd7 15. gxh5 Rxh5 16. Rhg1 Nxd4 17. Bxd4 Qc7 18. Rxg6 fxg6 19. Qxe6 Rf5 20. Nxd5 Qd8 21. Qxg6+ Rf7 22. Re1 Kf8 23. Qh7 Ke8 24. Qg8+ Rf8 25. Qxg7 Rf7 26. Qg8+ Rf8 27. Nf6# 1-0
It may not strike you right away, but the knight that is administering checkmate to the black king cannot be taken by three different pieces, because all three of them are pinned. The king’s only potential “escape square” is attacked only once, and the black pieces that are attacked and surround the black king are all pinned, so therefore this is a pure mate. And as you can see, multiple pieces are elsewhere on the board in the final position.
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!