Your analysis seems to assume that the only form of motivation is external. By this logic, persistent/obstinant people do things because they hope for some external reward (e.g., praise, recognition, fame, possibly financial compensation). The difference is merely at the granularity of the goal this is attached to.

My experience is that the persistent people I know have at least some degree (and often a large degree) of internal motivation. They do things because the process of problem solving is rewarding in and of itself, and/or they have some intrinsic motivation about solving the problem. They are not out to please anyone else except themselves.

Maybe no one is purely 100% internally motivated. But my experience is that the more persistent people I know generally have a higher percentage of internal motivation. In contrast the people who give up more easily generally have a lower percentage of internal motivation; if they really only care about the external reward, it often turns out there are lots of ways to do that, and many are shorter than solving "hard" problems.