Both these rules are ancient historical rules which were used, at one time or another, for the game of Shatranj. My reason for rejecting "Bare King" was that I saw interesting possibilities in the endgame King versus King, with no other pieces.
At first sight, it would seem that White to play is in zugzwang.
The move 1. Ke4-e5 allows 1...Ke6 flips to capture mode, checkmate.
The move 1. Ke4 turns towards d5 (or f5) allows 1...Ke6-e5, and because the path from d5 to d6 to e7 to e8 requires two turning moves, and the path from e5 to e1 is straight, Black wins the race, Black occupies the enemy throne first by one tempo.
White is forced to play 1. Ke4 flips.
In response, Black either flips or turns, and 2. Ke4 flips!
The move 2. Ke4 flips! is a clever move which was very difficult for me to find. The point is that repetition of position is forbidden, and so Black's King cannot undo its first move; and therefore Black cannot legally play the only move that would save the game.
If Black's first move was 1...Ke6 turns, then 2...Ke6 moves loses to 3. Ke4-e5; and so if Black's first move turned, Black's second move must flip, and vice versa. Thus, with White to play its third move, the WK at e4 is aimed straight forward and is flipped to movement mode, while the Black K is turned to d5 or f5 and is flipped to capture mode.
White's third move cannot be to move the K, because then Black would turn straight forward and that would be checkmate.
Instead, White must turn. It doesn't really matter which direction it turns.
For Black's third move, since the K is in capture mode it cannot move. One must make a move on the board at least once in every 5 turns, so Black must flip either now or next turn; but it doesn't matter, for if 3.... Ke6 flips, then 4...Ke6 turns; and vice versa.
White's 4th move is to move the King, whether Ke4-d5 or Ke4-f5 does not matter -- only one is legal, of course, it depends on which direction it turned on the third move.
Black's 4th is to flip or turn, as explained by a previous paragraph;
White's 5th turns straight forward, Black's 5th moves Ke6-e5,
then 6. Kf5-f6 or Kd5-d6 (whichever is legal), Ke5-e4,
7. Kd6-d7 or Kf6-f7, Ke4-e3,
8. K turns towards e8, Ke3-e2
9. K occupies enemy throne, end of game, White wins by one tempo.
I think that this is amazingly complex for an endgame with just the two Kings on the board.
In the basic position, if Black moves first, White still wins because the White King starts out closer to the enemy throne.
You may enjoy working out the results.
1... Kb7 turns towards center 2. Kb5 turns to center, Kb7-c6 3. Kb5 flips! (turning to b6 and taking the route b5-b6-c7-d8-e8 loses by one tempo because two turns are required).
Black is in check, so 3...Kc6-d5 4. Kb5 flips, Kd5-e4 5. Kb5-c6, Ke4 turns towards e3 6. Kc6-d7 and wins.
Knowledge of the "Case 1" basic position will allow you to calculate the result correctly. In order to play this game with the level of skill expected of a chessmaster, you are required to know the result long before you trade the last piece and get into a K versus K endgame; and it appears to me that this will sometimes involve thinking more than 30 moves ahead.
Did I say 30 moves? I am merely a master, not an IM nor and IGM, and yet, in tournament games of FIDE Chess, more than once I have calculated above 20 moves in advance in simple positions where exchanging the last pieces would lead to a King and Pawn endgame; which leads me to think that in a serious and slow game of Weakest Chess, an ordinary master (such as me) might well calculate 40 or 50 moves ahead. It is possible because the number of choices is so limited.
In addition, this endgame demonstrates basic techniques which will help you understand better how to handle the pieces in Weakest Chess; and in fact, this endgame is also valuable for giving you a feel for what the full game is like -- you should be able to tell from this endgame whether or not you would enjoy playing Weakest Chess; and this is a good thing to know because I suspect that many players will not enjoy the game at all but that many others will love it greatly.
The saga of Weakest Chess is far from finished.