Okay, I misunderstood, I did not consider the mathematical justification.
I just thought that R + gamma * q(S', A', w) - q(S, A, w) is about moving the weights of q(S, A, w) towards R + gamma * q(S', A', w), and that implementing it with any loss function that encoded this difference would do a similar thing.
I don’t quite follow how:
(1/2) * Q_w^2 - rQ_w - yQ'_w + C

can turn into:
(1/2) * (Q_w - (r + yQ'_w))^2 + C