We investigate the quantum phases of higher-spin Kitaev models using tensor network methods. Our results reveal distinct bond-ordered phases for spin-1, spin-$\frac{3}{2}$, and spin-2 models. In all cases, we find translational symmetry breaking with unit cells being tripled by forming valence-bond orders. However, these three phases are distinct, forming plaquette order, topological dimer order, and non-topological dimer order, respectively. Our findings are based on a cross-validation between variational two-dimensional tensor network calculations: an unrestricted exploration of symmetry-broken states versus the detection of symmetry breaking from cat-state behavior in symmetry-restricted states. The origin of different orders can also be understood from a theoretical analysis. Our work sheds light upon the interplay between topological and symmetry-breaking orders as well as their detection via tensor networks.