If someone is a true determinist, then they hold for some variant of eternalism, the future is no different to the present and the past, modifying it is meaningless. That does not imply they act without self-motivation or agency, the external in your question assumes they are separate from the rest of the univers, you can solve that by understanding they are as much driven toward their actions as would a ball move in compliance to the physical laws, they will to act as much as the ball wills to move, but nothing grants you the right to assume the ball is moved by an external force (except for a physical representation of the mechanic laws), you might as well believe the ball actively wills to move in accordance with the gravity force direction. In this sense, the logic to decide is the logic of existence, your Will equates your Nature exactly, no more no less.
The questions remains however: why is it possible then to imagine different natural outcomes? I would think a determinist will answer by refuting that such an act of imagination is either impossible (you could draw a penrose triangle but never actualize it) or irrelevant (before a ball moves, it might go everywhere, except it only could/will go in a specific direction);
The take-away is this: if you want to understand determinism, you've got really to understand it's deterministic, being a determinist while keeping with extra-deterministic ideas does not pan out. You could also just reject it and that's fine.