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Relation between Altmann’s definition of semi-direct product and the one in Wikipedia

First off, notice that Altmann’s says that $$\mathbf{S}$$ (the set of coset representatives) is a group. Secondly, $$\mathbf{H}\triangleleft\mathbf{S}$$ is equivalent (as can be easiliy shown) to being a normal subgroup of $$\mathbf{G}$$.
Thus, $$\mathbf{H}$$ is a normal subgroup of $$\mathbf{G}$$. Therefore, we recover the original meaning of $$\mathbf{G} = \mathbf{H} \rtimes \mathbf{S}$$.