I think the simplified form of loss function depends on y^{(i)} but not f_{\mathbf{\vec{w}},b}(\mathbf{x}^{(i)}). So the f_{\mathbf{\vec{w}},b}(\mathbf{x}^{(i)}) in question and the explanation should be y^{(i)}.

$$y^(i)$$ is the target. The loss function offers two scenarios, one is when the model’s predicted value is 1, and the other is 0. The target is the ground truth, can not be set by the model. The question here is saying that the model’s predicted value is 1. So the simplified expression is correct.