Current Value Hamiltonian VS Present Value Hamiltonian in Economics
There is no clear right and wrong about this, it's just a matter of convenience. The current-value Hamiltonian is likely to be more convenient when the objective function includes a discount factor. Following Chiang (1), suppose the problem is:
$\qquad$Maximise $V = \int_0^T G(t,y,u)e^{-\rho t}$
$\qquad$subject to $\dot y=f(t,y,u)$
$\qquad$and boundary conditions
The standard (present value) Hamiltonian is:
$\qquad H=G(t,y,u)e^{-\rho t} + \lambda f(t,y,u)$
If we proceed from this Hamiltonian, the co-state equation (one of the first-order conditions) is:
$\qquad \dot \lambda = -\frac{\partial H}{\partial y}= -\frac{\partial [G(.)e^{-\rho t}]}{\partial y}-\lambda\frac{\partial f}{\partial y}$
While it is possible to obtain a solution this way, the discount factor complicates the derivatives and can make interpretation more challenging.
Suppose instead we use the current-value Hamiltonian:
$\qquad H_c = G(t,y,u) + mf(t,y,u)$
where $m$ is a current-value Lagrange multiplier defined by $m=\lambda e^{\rho t}$.  The co-state equation is then:
$\qquad \dot m -\rho m = -\frac{\partial H_c}{\partial y} = -\frac{\partial G}{\partial y} - m\frac{\partial f}{\partial y}$
This is simpler because it contains no discount term.
Reference

Chiang A C (1992) Elements of Dynamic Optimization pp 210 ff