"rumor" vs "unfounded rumor"

If I'm not wrong, "rumor" means:

A story or report of uncertain or doubtful truth

and "unfounded" means:

Having no foundation or basis in fact

If so, what is the difference between an "unfounded rumor" and a "rumor"?

Solution 1:

A rumor might be true or untrue. Even if it's untrue, though, it might be based on fact:

Fact: Alice and Bob have been spending a lot of time together.
Rumor: Alice and Bob are going steady.
Reality: Alice and Bob are not dating, they are partners for a school project.

In contrast, an unfounded rumor would be based on no credible supporting evidence:

"I think Alice and Bob are going out!"
"Really? Why do you say that?"
"I saw them talking in the hallway."

(Alice and Bob could have been talking about anything; Alice may have simply dropped a pencil, and Bob was handing it back to her. They may not even know each other's names! That looks like the birth of an unfounded rumor.)

Solution 2:

A rumor is uncertain or doubtful (but may be true). An unfounded rumor has no foundation or basis in fact (and thus cannot be true).