In many routing problems, it is fairly common to include a constraint that ensures all vehicles follow an elementary path, meaning that no vertices are repeated.
However, when an elementary path is NOT required, then it is fairly common to include a constraint that ensures there are no subtours meaning that there are no isolated tours that do not start at, and return to, the specified depot.
From my understanding, it would never be necessary to include both of these constraints. However, in some cases I have seen both of these constraints, for example here: https://par.nsf.gov/servlets/purl/10074741
EDIT I've found another resource which highlights my intuition: In this book it describes a similar problem, the vehicle routing problem with time windows. And, in its MILP formulation (shown below), it does not have subtour elimination constraints, and explicitly mentions the classical VRP subtour elimination constraints become redundant. Now, I would have expected this to be the same reason why the image 2 would not need the subtour elimination constraints, since in both cases, the direction of traversal is indicated by denoting $(i,j)$.
The difference between these two MILP formulations and problem statements might help me understand what you guys are getting at. Is the only difference that one of them imposes a unique direction?