According to recent work of Greensite and Matsuyama, the Higgs phase of a gauge-Higgs theory is distinguished from the confinement and massless phases by the spontaneous breaking of a global center subgroup of the gauge group, and by confinement type. This is contrary to the notion that there is no essential distinction between the Higgs and confinement phases when the Higgs field is in the fundamental representation of the gauge group. Although this new symmetry-breaking order parameter has been investigated in $D=4$ dimensions, there is so far no check in a non-Abelian gauge theory containing a massless as well as confinement/Higgs phases, where the prediction is that the symmetry-breaking order parameter will show transition lines separating the massless-to-Higgs and confinement-to-Higgs phases, but not the massless-to-confinement phase. In this work we map out the phase structure of the $D=5$ dimensional model, according to both the symmetry-breaking parameter and thermodynamic observables, and check the assertion regarding the massless-to-confinement phase.