Why think these are the same thing?

They are not the same thing, but they aren't independent. And they are not only causally dependent, but logically - which is why CDT intervention at the action node, leaving the agent-type node untouched, makes no sense. CDT behaves as if it were possible to be one agent type for the purpose of Omega's prediction, and then take an action corresponding to another agent type, even though that is logically impossible. CDT is unable to view its own action as predetermined, but its action is predetermined by the algorithm that is the agent. TDT can take this into account and reason with it, which is why it's such a beautiful idea.