Even your link doesn't refute the claim.
As a case in point, thinking the Klan a patriotic organization, future president Harry S. Truman, a Mason and an up-and-coming politician running for county office in the late summer of 1922, was ready to join the order. Indeed, he had given his $10.00 to an organizer and he may even have taken the Klan oath and signed a membership card.
https://www.coolidgefoundation.org/blog/the-ku-klux-klan-in-calvin-coolidges-america/