The loop is the mechanism. The agent is the thing that owns it and survives between runs.
The loop is the mechanism: one run, from trigger to commit. The agent is the thing that owns the loop and survives between runs. The loop runs and ends. The agent persists, and it is the agent, not the loop, that has a name, a seat, tools, a memory and a budget.
One agent, three runs. The ring is the same code every time. The agent is what makes those three runs "Oddz" and not "anybody".
Test for the word: if it can wake up tomorrow and still be itself, it is an agent. If it starts, does its passes and is gone, that was a run of the loop.