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There are at least two ways for white to mate within four moves, beginning either Nxg7 or Rxg7, but those moves are not accepted as the "right" move. The "right" move depends on the black queen moving into jeopardy for no good reason.

2. Nxg7 - Nxg7
3. Qxh6.checkmate
OR
2. Rxg7 - Rxg7
3. Kxg7 - Kxg7
4. Qxh6+ - Kg8
5. Rg1 - ...moot move
6. Qxg7.checkmate

Okay, one option was mate in two and the other is mate in five, but who's counting? Either play is more reasonable than the official solution.

Sorry, I disagree. In each of your examples, the correct Black move is Rxg7 since the Black King now has an escape square to g8. The comment that Black's fifth move is 'moot' does not apply, since the king can move again, as in 2. Rxg7 Rxg7 3. Nxg7 Nxg7 4. Qxh6+ Kg8 5. Rg1 Kf8 6. Qxg7+ Ke7 7. Qf6+ Kd7 and White is hard put to mate soon.

I think the solution of 2. Nxh6 is right on the money since the threat of mate is imminent.
If Black tries 2. Nxh6 gxh6 3. Qxh6#. Or, 2. Nxh6 Ng5 3. Nxf7#. Or, 2. Nxh6 Rf8 3. Nxf7+ Kg8 4. Nxd8 grabs the Queen.