I like this idea since most of the best chess variants are just a normal chess game but with a twist. In a way it's like the opposite of chess960 (where the opening game is radically different but endgame behave as usual) you wouldn't notice the new rule at first as the King is tucked away, but in endgame it could make a big difference
DeFree Chess
I don't think these are consistent rules. You would need a whole lot of additional rules to unambiguously specify what is allowed and what not. In the diagram above you could again add a white Bishop or Queen on c8. It is also not clear whether you could step with your own King next to the opponent's King. If your own King is protected, this is logically the same situation as a pin: he could not capture your King with his King without exposing it to capture by the protector.
To me the defree rule is fundamentally illogical. The idea of chess is that as soon as one of the Kings gets captured the game is over. So a pinned piece can check the King because it could capture that King and end the game before the piece that was pinning it can capture the other King.

@p-wnattack: no, white is not in check there, the rule depth is 1 turn only, it stops at the pins on moving player and by the moving player.
@HGMuller: I think the 1 turn depth is the reply to the queen/bishop in c8 too. As for the two kings, I haven't thought about it until now, but it's enough to apply the rule straightforward. So yes, it can step next to the opponent if that square is protected, but then the piece protecting that square must keep protecting it and, if it falls or if its way to the king gets blocked, then the king becomes in check. For example:
Here the white can play Kf3+. Then black king is in check, but white is not. Black cannot play Rxb7++, because black king is already in check and must escape first. Likewise, black cannot play d5++. There's no legal move that can make both kings check I think, because, apart from the pinning rule, other chess rules are still valid, so the each player can only play moves that make them escape the check position.
In the following case Kf3 is possible and neither king is in check. However black bishop must keep protecting his king and so does white. Black could play d5+ for example. Or, if black played be6, white couldn't play d5-+ afterwards, because that would set his own king in check.
Even if you found a position and a move that breaks two pins in one shot and leads to both kings in check, then that move is not legal, because it makes your own king in check (following the standard chess rules).
@Casual_Joe: you have a point, but I think that, in the end, it's only a matter of agreeing upon some rules. It's obviously illogical under standard chess rules, but here I'm suggesting a new variant, e.g. new rules. In DeFree, it isn't enough to capture the other king, you must also ensure that your own king is still protected from the opponent pieces. Should you dislike that, you can always play something else.
Which stands for "Degrees of Freedom constrained Chess".
This chess variant adds a simple rule to standard chess (or to any other chess variant you like): a piece does not check the opponent's king if it cannot move from its square.
Consider the following position, black to move:
In standard chess, this is a checkmate, 1-0.
In DeFree chess, the black king is allowed to take white queen, because the bishop in g3 cannot actually move: if it moved, the white king would be in check.
In standard chess the bishop in g3 cannot defend the queen until the white king moves away from the opponent's queen file. In DeFree chess it cannot even check the opponent's king for the same reason.
Here is another scenario where DeFree chess offers more tactical tricks than standard chess (white to move, please note that black is not under check because c2 bishop cannot move):
Here the mate in 1 is Raxh7#, because that frees up the c2 bishop, which now checks the opponent's king.
Another interesting position:
Here the queen is not attacking the king because of the bishop. Likewise, the white rook is not attacking the king because of the black rook.
I don't know if this chess variant had already been invented: if yes, please tell me its name, otherwise, please tell me what you think about it.