It has been proposed that motivating participants to perform well on a cognitive task ought to lead to decreases in rates of intentional, but not unintentional, task-unrelated thought (TUT; a commonly studied variety of mind wandering). However, at odds with this prediction, research has found that increasing motivation results in decreases in both intentional and unintentional TUTs. One possible explanation for this surprising finding is that standard assessments of TUT may inadvertently conflate TUTs with another variety of mind wandering: unconstrained thought. If so, then deconfounding task-unrelated and unconstrained varieties of mind wandering might produce the predicted effect of a decrease in intentional, but not unintentional, TUT when motivation is increased. To explore this possibility, in the present study, participants completed a sustained-attention task after receiving standard instructions (normal-motivation condition) or instructions informing them that they could leave the study early if they achieved a certain level of performance (motivated condition). Throughout the task, we assessed rates of TUT (both intentional and unintentional) and unconstrained thoughts. Consistent with prior work, the results indicated that motivated participants reported being on-task significantly more frequently than non-motivated participants. However, unlike previous work, we found that when deconfounding TUTs and unconstrained thoughts, participants in the motivation condition reported significantly fewer bouts of intentional TUT than those in the non-motivation condition, but no differences in rates of unintentional TUT were observed between groups. These results suggest that (a) motivation specifically targets intentional TUT and (b) standard assessments of TUT conflate task-relatedness and thought constraint.