To fully understand the difference between get_categories(), get_category() and get_term_by(), you need to have a look at how these functions are constructed.
get_categories($args) as its name suggest, get a list of all categories created on the site. This function can be found in “wp-includes/category.php” lines 39 to 66
39 function get_categories( $args = '' ) {
40 $defaults = array( 'taxonomy' => 'category' );
41 $args = wp_parse_args( $args, $defaults );
42
43 $taxonomy = $args['taxonomy'];
44 /**
45 * Filter the taxonomy used to retrieve terms when calling get_categories().
46 *
47 * @since 2.7.0
48 *
49 * @param string $taxonomy Taxonomy to retrieve terms from.
50 * @param array $args An array of arguments. @see get_terms()
51 */
52 $taxonomy = apply_filters( 'get_categories_taxonomy', $taxonomy, $args );
53
54 // Back compat
55 if ( isset($args['type']) && 'link' == $args['type'] ) {
56 _deprecated_argument( __FUNCTION__, '3.0', '' );
57 $taxonomy = $args['taxonomy'] = 'link_category';
58 }
59
60 $categories = (array) get_terms( $taxonomy, $args );
61
62 foreach ( array_keys( $categories ) as $k )
63 _make_cat_compat( $categories[$k] );
64
65 return $categories;
66 }
As you can see from line 40, category is used as the default taxonomy, so only categories will be returned in the list of categories
get_category( $category, $output, $filter ) only retrieves the category data for a specific given category. This function can be found in “wp-includes/category.php” lines 90 to 99
90 function get_category( $category, $output = OBJECT, $filter = 'raw' ) {
91 $category = get_term( $category, 'category', $output, $filter );
92
93 if ( is_wp_error( $category ) )
94 return $category;
95
96 _make_cat_compat( $category );
97
98 return $category;
99 }
From line 91, you can see that get_category() only uses category as the term, so only category will be returned, no terms.
get_term_by( $field, $value, $taxonomy, $output, $filter ) will get all term data by term field and data. This function can be found “wp-includes/taxonomy.php”in lines 1028 to 1076
1028 function get_term_by($field, $value, $taxonomy, $output = OBJECT, $filter = 'raw') {
1029 global $wpdb;
1030
1031 if ( ! taxonomy_exists($taxonomy) )
1032 return false;
1033
1034 if ( 'slug' == $field ) {
1035 $field = 't.slug';
1036 $value = sanitize_title($value);
1037 if ( empty($value) )
1038 return false;
1039 } else if ( 'name' == $field ) {
1040 // Assume already escaped
1041 $value = wp_unslash($value);
1042 $field = 't.name';
1043 } else if ( 'term_taxonomy_id' == $field ) {
1044 $value = (int) $value;
1045 $field = 'tt.term_taxonomy_id';
1046 } else {
1047 $term = get_term( (int) $value, $taxonomy, $output, $filter);
1048 if ( is_wp_error( $term ) )
1049 $term = false;
1050 return $term;
1051 }
1052
1053 $term = $wpdb->get_row( $wpdb->prepare( "SELECT t.*, tt.* FROM $wpdb->terms AS t INNER JOIN $wpdb->term_taxonomy AS tt ON t.term_id = tt.term_id WHERE tt.taxonomy = %s AND $field = %s LIMIT 1", $taxonomy, $value) );
1054 if ( !$term )
1055 return false;
1056
1057 wp_cache_add($term->term_id, $term, $taxonomy);
1058
1059 /** This filter is documented in wp-includes/taxonomy.php */
1060 $term = apply_filters( 'get_term', $term, $taxonomy );
1061
1062 /** This filter is documented in wp-includes/taxonomy.php */
1063 $term = apply_filters( "get_$taxonomy", $term, $taxonomy );
1064
1065 $term = sanitize_term($term, $taxonomy, $filter);
1066
1067 if ( $output == OBJECT ) {
1068 return $term;
1069 } elseif ( $output == ARRAY_A ) {
1070 return get_object_vars($term);
1071 } elseif ( $output == ARRAY_N ) {
1072 return array_values(get_object_vars($term));
1073 } else {
1074 return $term;
1075 }
1076 }
In line 1047, you can see that the taxonomy can be specified by the user, and is not harcoded than the other two functions previously mentioned
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