Loop gain(s) for multi-loop circuits
The diagram can be rearranged so that \$H_2G_2\$ is an inner loop and $$\frac{G_1H_1G_2}{1+G_2H_2}$$ is the outer loop.

It is required for the outer loop to be stable. The inner loop may be unstable with the full loop being stabilised by controller \$G_1\$. This is generally only done when absolutely necessary to stabilise an unstable plant.

Practically speaking, both loops should be required to be stable in a system such as this with standard op-amps.

As a rule of thumb for design simplicity, the \$G_2H_2\$ loop is designed to be stable with significantly higher bandwidth than the final loop bandwidth so that its phase delay can be neglected in the design of the outer loop; i.e. its phase/magnitude contribution to the overall loop is negligible at crossover.



To the OP's Update: The point of which loop to analyse: A straightforward analysis of the OP's loop (also the Updated equivalent arrangement) shows the stability analysis to be:
$$G_2(G_1H_1+H_2)+1=0$$

for the rearranged circuit with the inner loop we get:
$$\frac{G_1G_2H_1}{1+G_2H_2}+1=0$$

Applying simple algebra we see that it's the same result, as expected:
$$G_2(G_1H_1+H_2)+1=0$$

Therefore you can analyse the loop several ways and get the same result. Considering the loop to consist of an inner and outer loop, is however, very convenient.

UPDATE:
Here's the analytic justification for the loop stability equation to be $$G_2(G_1H_1+H_2)+1=0$$ Apologies for quality. I hope to enter as native format when I get time: