The weak equivalences in the model structure that $* \downarrow \mathbf{Top}$ inherits formally are pointed maps $f: X\to Y$ that are unpointed homotopy equivalences; but in his discussion of the pointed case, Str$\varnothing$m Strøm takes $\mathbf{Top}_*$ to have pointed homotopy equivalences as weak equivalences.
The weak equivalences in the model structure that $* \downarrow \mathbf{Top}$ inherits formally are pointed maps $f: X\to Y$ that are unpointed homotopy equivalences; but in his discussion of the pointed case, Str$\varnothing$m takes $\mathbf{Top}_*$ to have pointed homotopy equivalences as weak equivalences.