Semiflexible polymers are widely used as a paradigm for understanding structural phases in biomolecules including protein folding. Here, we compare bead-spring and bead-stick variants of coarse-grained semiflexible polymer models that cover the whole range from flexible to stiff by conducting extensive replicaexchange Monte Carlo computer simulations. In the data analysis we focus on knotted conformations whose stability is shown to depend on the ratio rb/rmin where rb is the equilibrium bond length and rmin the distance of the strongest nonbonded attractive interaction. For both models, our results provide evidence that at low temperatures for rb/rmin outside a small range around unity one always encounters knots as generic stable phases along with the usual frozen and bent-like structures. By varying the bending stiffness, we observe rather strong first-order-like structural transitions between the coexisting phases characterized by these geometrically different motifs. Through analyses of the energy distributions close to the transition point, we present exploratory estimates of the free-energy barriers between the coexisting phases.