Use a later hook to tweak the options.

controlled_access_terms.field_type_categories.yml

@@ -1,6 +1,6 @@
 typed_relation:
   label: 'Typed Relation'
   description: 'A Reference field, with an associated type selected from a configurable list.'
-  weight: 10
+  weight: 100
   libraries:
     - controlled_access_terms/controlled_access_terms.icons

controlled_access_terms.libraries.yml

@@ -1,4 +1,4 @@
 controlled_access_terms.icons:
   css:
     theme:
-      css/controlled_access_terms.icons-min.css: {}
+      css/controlled_access_terms.icons.css: {}

controlled_access_terms.module

@@ -7,6 +7,7 @@
 
 use Drupal\jsonld\Normalizer\NormalizerBase;
 use Drupal\Core\Entity\EntityInterface;
+use Drupal\Core\Field\FieldTypeCategoryManagerInterface;
 use Drupal\controlled_access_terms\EDTFConverter;
 use Drupal\field\Entity\FieldStorageConfig;
 use Drupal\field\Entity\FieldConfig;
@@ -111,6 +112,27 @@ function controlled_access_terms_form_field_ui_field_storage_add_form_alter(&$fo
   $form['#attached']['library'][] = 'controlled_access_terms/controlled_access_terms.icons';
 }
 
+/**
+ * Implements hook_field_info_entity_type_ui_definitions_alter().
+ */
+function controlled_access_terms_field_info_entity_type_ui_definitions_alter(array &$ui_definitions, string $entity_type_id) {
+  // Put the generic typed relation field last and name it 'Other'.
+  if (isset($ui_definitions['typed_relation'])) {
+    $ui_definitions['typed_relation']['label'] = t('Other');
+    $ui_definitions['typed_relation']['weight'] = 10;
+  }
+  // Undo some shenanigans done by the Media module to the media option.
+  if (isset($ui_definitions['field_ui:typed_relation:media'])) {
+    $ui_definitions['field_ui:typed_relation:media']['category'] = 'typed_relation';
+    $ui_definitions['field_ui:typed_relation:media']['description'] = 'A reference field with a configurable type selector.';
+    $ui_definitions['field_ui:typed_relation:media']['weight'] = 0;
+  }
+  // Put taxonomy term first in the option list.
+  if (isset($ui_definitions['field_ui:typed_relation:taxonomy_term'])) {
+    $ui_definitions['field_ui:typed_relation:taxonomy_term']['weight'] = -10;
+  }
+}
+
 /**
  * Update EDTF fields from the 2012 draft to match the 2018 spec.
  */

src/Plugin/Field/FieldType/TypedRelation.php

@@ -15,7 +15,7 @@
  *   label = @Translation("Typed Relation"),
  *   module = "controlled_access_terms",
  *   category = "typed_relation",
- *   description = @Translation("Implements a typed relation field"),
+ *   description = @Translation("A reference field with a configurable type selector."),
  *   default_formatter = "typed_relation_default",
  *   default_widget = "typed_relation_default",
  *   list_class = "\Drupal\Core\Field\EntityReferenceFieldItemList",