MySQL varchar index length

I have a table like this:

CREATE TABLE `products` (
  `id` int(11) unsigned NOT NULL AUTO_INCREMENT,
  `name` varchar(512) NOT NULL,
  `description` text,
  PRIMARY KEY (`id`),
) ENGINE=InnoDB AUTO_INCREMENT=38 DEFAULT CHARSET=utf8;

and one like this:

CREATE TABLE `product_variants` (
  `id` int(11) unsigned NOT NULL AUTO_INCREMENT,
  `product_id` int(11) unsigned NOT NULL,
  `product_code` varchar(255) NOT NULL,
  PRIMARY KEY (`id`),
  UNIQUE KEY `product_code` (`product_code`),
  KEY `product_variant_product_fk` (`product_id`),
  CONSTRAINT `product_variant_product_fk` FOREIGN KEY (`product_id`) REFERENCES `products` (`id`)
) ENGINE=InnoDB AUTO_INCREMENT=1037 DEFAULT CHARSET=utf8;

and an SQL statement like this

SELECT p.id AS id, p.name AS name, p.description AS description, pv.id AS product_variant_id, pv.product_code AS product_code
FROM products p
INNER JOIN product_variants pv ON pv.product_id = p.id
ORDER BY p.name ASC
LIMIT 300 OFFSET 0;

which if I explain gives me this:

+----+-------------+-------+------+----------------------------+----------------------------+---------+---------+--------+----------------+
| id | select_type | table | type | possible_keys              | key                        | key_len | ref     | rows   | Extra          |
+----+-------------+-------+------+----------------------------+----------------------------+---------+---------+--------+----------------+
|  1 | SIMPLE      | p     | ALL  | PRIMARY                    | NULL                       | NULL    | NULL    | 993658 | Using filesort |
|  1 | SIMPLE      | pv    | ref  | product_variant_product_fk | product_variant_product_fk | 4       | db.p.id |      1 |                |
+----+-------------+-------+------+----------------------------+----------------------------+---------+---------+--------+----------------+
2 rows in set (0.00 sec)

Fo