How to set “manage categories” capabilities on a post type taxonomy but not on the general categories?

you can just define your own capabilities to use this taxonomie

    'manage_terms'          => 'manage_terms_TAXONOMIE_NAME',
    'edit_terms'            => 'edit_terms_TAXONOMIE_NAME',
    'delete_terms'          => 'delete_terms_TAXONOMIE_NAME',
    'assign_terms'          => 'assign_terms_TAXONOMIE_NAME',

and after you choose wich capabilities you give to your users

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