The word "extortion," like the words blackmail and fraud, are often used casually by laypersons. And that's okay. Laypersons are not expected to know or understand legal terminology. On a legal forum, however, most of the forum members are sensitive to the legal meanings and most of the forum advisers will use the words as they are defined legally.
I think you are very smart to try to negotiate a reasonable price to pay the designer in exchange for all rights in the logo. It is unfortunate that the designer is demanding payment now because a dispute with the club has made her decide to assert IP rights, but it is what it is.
Yours is indeed a good cautionary tale. Agreements should always be in writing and dated and signed by the parties. Disputes are not uncommon. A written agreement can work to eliminate the source of many of them.
Any update you wish to provide would be welcomed. Good luck.