Thanks both for your replies.
Regarding Johannes’ post, what you say indeed explains the difference in dynare solution between the two versions. And it also shows why I think it is a suboptimal timing convention.
In my opinion, the first constraint in the Lagrangian should be read as (I added the expectations, similarly I could add them in the second constraint)
Since E_0(x_1)~= x_1 (since the shock at t=1 is an MIT shock) the ancient value r_0 should hence not be relevant for the second constraint.