Index: src/wp-includes/rest-api/endpoints/class-wp-rest-terms-controller.php
===================================================================
--- src/wp-includes/rest-api/endpoints/class-wp-rest-terms-controller.php	(revision 39362)
+++ src/wp-includes/rest-api/endpoints/class-wp-rest-terms-controller.php	(working copy)
@@ -349,7 +349,7 @@
 		}
 
 		$taxonomy_obj = get_taxonomy( $this->taxonomy );
-		if ( ! current_user_can( $taxonomy_obj->cap->manage_terms ) ) {
+		if ( ! current_user_can( $taxonomy_obj->cap->edit_terms ) ) {
 			return new WP_Error( 'rest_cannot_create', __( 'Sorry, you are not allowed to create new terms.' ), array( 'status' => rest_authorization_required_code() ) );
 		}