Structural damages result in nonlinear dynamical signatures that significantly help for their monitoring. A damage type classification approach is proposed here that is based on a parallel Hammerstein models representation of the structure estimated by means of the Exponential Sine Sweep Method. This estimation method has been here extended to take into account for input signal amplitude which was not the case before. On the basis of these estimated models, three amplitude dependent damage indexes are built: one that monitors the shift of the resonance frequency of the structure, another the ratio of nonlinear versus linear energy in the output signal, and a last one the ratio of the energy coming from odd nonlinearities to the energy coming from even nonlinearities in the output signal. The slopes of these amplitude-dependent DIs are then used as coordinates to place the damaged structure under study within a three-dimensional space. A single mass-spring-damper system is considered to illustrate the ability of this space to classify different types of damage. Four types of damage with different severities are simulated through different spring nonlinearities: bilinear stiffness, dead zone, saturation, and Coulomb friction. For all severities, the four types of damage are extremely well separated within the proposed three-dimensional space, thus highlighting its high potential for classification purposes.