I have one specific equation that if it written as

L = L(+1) + r - pi(+1)

the model is not solved (Indeterminacy)

However, if I iterate this equation one period back, so that

L = L(-1) - r(-1) + pi

the model can be solved. Both equations should imply the same information for a RE equilibrium, and both equations imply the same forecasting rule. Do you have any explanation as to the reasons why two equations differ in the sense of model determinacy? Thanks in advance.

Your premise is wrong: both models are different. It is difficult to discuss on the basis of this example, because the model is incomplete. The first equation states that L_t depends on the conditional expectations\mathbb E_t [L_{t+1}] and \mathbb E_t [\pi_{t+1}], while the second equation says that L_t depends on past level of L and the actual realisation of \pi in period t. You cannot solve for the expectations by simply shifting backward an equation, ie by implicitly replacing expectations by actual realisations, doing so you totally ignore the expectations. Note also that if such a transformation was legal, we could apply it to all models and we won’t need to develop special algorithms (or create Dynare) to solve them

Obviously, it is possible to shift backward an equation, but you would also have to shift the information sets used for the expectations, so you would have:

but I do not think you will find this transformation very useful…

On top of that, you have to be aware that Dynare identifies the state variables vs non predetermined variables thanks to the timing convention. Variables in t-1 are state variables. So when you artificially shift backward an equation, you mess up the counting of the state variables and the determinacy/stability test.