I am given a smooth connected manifold $M$ with two smooth metric tensors $g_1$ and $g_2$. Let us denote by $d_1$ and $d_2$ the distance they induce on $M$, respectively. I have two ways of "mixing" these two distances on $M$: for points $z_0, z_1 \in M$ we can define
$$d'(z_0,z_1)^2:= \inf \left \{ \int_0^1 \left (|v_1(t)|^2_{g_1(x(t))} +|v_2(t)|^2_{g_2(x(t))}\right ) d t : x \in C^1([0,1];M), x(0)=z_0, \, x(1)=z_1,\, \dot{x}(t)=v_1(t)+v_2(t) \right \}$$
and
$$ d''(z_0,z_1)^2:= \liminf_{N \to + \infty} \inf \left \{ N\sum_{i=1}^N \left ( d_1(x_{i-1},y_i)^2 + d_2(y_i,x_i)^2 \right ) : z_0=x_0,y_1, x_1, \dots, y_N, x_N=z_1 \in M \right \}. $$
My question is: are these two definitions equivalent under reasonable assumptions? How can we prove it?