Model types: Robust diagnostic versus prognostic
As far as I know in the ocean modeling community the terms diagnostic and prognostic have quite different meanings from what you are stating. 

A prognostic simulation is used to predict future state of the system (forecast) using the model equations. The model is initialized with the initial conditions and it is used to predict using the Navier-Stokes equations the future ocean state under specified boundary conditions. There is no constraint by observations because the model numerically follows the specified equations. In the case of models with data assimilation, the prognostic solution is statistically modified (e.g., Kalman filter, adjoint method) to optimally adjust to the observations, but the new solution can't be considered prognostic as there has been a departure from the primitive forward (Navier-Stokes) equations.

On the other hand, a diagnostic simulation does not provide a prediction of the fluid state. There is no time-evolution based on the model equations. An example of such simulation is the case in which the velocity field is calculated based on a fixed density (temperature and salinity) field. While these simulations can be useful for short-term ocean state estimation, they can't be used for ocean state evolution as the dynamics are restricted.

As far as I understand it, this is the standard in the ocean modeling community, but I don't know if it can be expanded to other fields.