Currently we interpret a story without tasks as not started.

On sprint planning every story must get at least one task. If the task later turns out to be done somehow, one would close the task without burning hours.

The distinction makes a rather technical impression to me.

Why should a Story that has Tasks that are all complete, be unfinished? Do we have a real reason to do it this way (like the case when Stories are reused for later Sprints)?

If not, I would suggest treating Stories as described in this issue as completed.

When pulling a story into a sprint, this story would be immediately displayed as completed. Showing the green hook and the accept/reject assistence box to the product owner.

I don't think that completing all planned tasks surely means that the story is complete. Sometimes the team misunderstands the goals of that story, or the product owner simply did not specify everything in that backlog.

In my oppinion, that's why accepting or rejecting stories should by done by product owner.

One quick note: If the Product Owner has incorrectly specified the Story in the Backlog, it's his fault. Whether a Story is accepted or rejected may only be based on the desctiption (more precisely, the acceptance criteria) given in the Story.

However, you are correct on everything else. But that does not conflict with this issue. "Completed" means that a Story is marked for review by the Product Owner. Nothing more.