Without having read the sources in detail, I would notice the following:

In (4) the second measure is time independent. To satisfy (1) this then corresponds to $(\rho^2_t,v_t^2) = (\nu,0)$. But then since $v_t^2$ vanishes, so does the second term in (2).

All the other changes are already noticed in the question. One is just the specific definition of the vector field. The other is the fact that for absolutely continuous measures the unique solution $T_t$ can be expressed as a gradient $\nabla \psi_t$, where $\psi_t$ is a convex function. This is a well known theorem (due to Brenier, I believe) which is found in any textbook on optimal transport.

Without having read the sources in detail, I would notice the following:

In (4) the second measure is time independent. To satisfy (1) this then corresponds to $(\rho^{(2)}_t,v_t^{(2)}) = (\nu,0)$. But then since $v_t^{(2)}$ vanishes, so does the second term in (2).

All the other changes are already noticed in the question. One is just the specific definition of the vector field. The other is the fact that for absolutely continuous measures the unique solution $T_t$ can be expressed as a gradient $\nabla \psi_t$, where $\psi_t$ is a convex function. This is a well known theorem (due to Brenier, I believe) which is found in any textbook on optimal transport.