wp_insert_post or wp_set_post_terms do not save taxonomy, but wp_set_post_terms does

Have you tried:

wp_set_object_terms( $postId, array( 'obrazek'), 'rodzaj' );

wp_set_object_terms and wp_set_post_terms take the same arguments. wp_set_post_terms even uses wp_set_object_terms internally. The main difference being that you used an array in the one that worked, and you didn’t use an array in the one that didn’t work.