About puzzle 37688

I was right at the 1st move: B: Kg5xBe4 - W: d3xe4

Puzzle solution :2) B: Qh8-h2+ - W: Kf2-f1
3) B: Qh2xd2
But what if ? 2)B: Qh8-h2+ - W: Kf2-e1?
Then best for black is :
3)Qh2xg1+ - Ke1-e2
4)Qg1xa1
White queen is saved for a bishop and a rook

Wouldnt it be better if ? 2)Rb8-b2 - Qd2Xb2
3)Qh8-h2+ Ke2-f1
4)Qh2xb2
and white loses the queen