There are actually three different versions of DR20 on the three cards that have it.

AT-AT Imperial Walker, the only WOTC character with the ability, has "Whenever this character takes damage from a nonadjacent enemy, reduce the damage dealt by 20. Enemies with lightsabers ignore this special ability." This version specifies that enemies with lightsabers ignore the ability, rather than attacks with lightsabers- which could actually matter for characters who do not have lightsabers, but have abilities that make their attacks count as attacks with lightsabers. Another curiosity is that the Walker also has Weak Spots 10, an ability that reduces its Damage Reduction to 10 against attacks from adjacent enemies- but its Damage Reduction already doesn't work against adjacent enemies. Or does it? Maybe the glossary takes precedence over the card text, or maybe being reduced to 10 means that it works the same way as DR10.

V-set 17's Fambaa has a version that works against adjacent enemies like DR10: "Whenever this character takes damage, reduce the damage dealt by 20. Attacks with lightsabers ignore this special ability". This was presumably an error- the character originally had DR10 in design, and then it was changed from memory without referencing bloomilk or the AT-AT's card.

V-set 21's Mudhorn has "Whenever this character takes damage from a nonadjacent enemy, reduce the damage dealt by 20. Attacks with lightsabers ignore this special ability." It returns to precedent with adjacency, but not to exactly the same wording as the AT-AT. And like the AT-AT, it has Weak Spots 10.

All that said, I don't actually know the answer as to how DR20 works.

The discrepancy between card text and the official glossary is a classic Star Wars Miniatures headache.

In Star Wars Miniatures, the Glossary (and Errata) typically takes precedence over card text to ensure game balance. Nickname's May 2010 Final & Complete WotC Glossary defines "Damage Reduction [#]" universally, it effectively "patches" the text on the cards.

Regardless of the card text on the AT-AT or Mudhorn, the ability functions as Damage Reduction 20 using the Glossary definition. This means it does work against adjacent enemies, unless they have a lightsaber.

If DR20 only worked against non-adjacent enemies (per the AT-AT card text), Weak Spots 10 would be useless.
The existence of Weak Spots 10 on the AT-AT and Mudhorn proves that the designers intended for DR20 to function against adjacent enemies.

"Enemies with Lightsabers" vs. "Attacks with Lightsabers"
The AT-AT's card text ("Enemies with lightsabers") is an early-set wording quirk. Per the 2010 Glossary, it is the attack that matters. If a character like Guri (who does not have a lightsaber) uses an ability that makes her attack count as a lightsaber attack, she ignores DR20. Conversely, if a Jedi makes an attack that does not count as a lightsaber attack (rare, but possible via certain Force powers), DR20 would apply.

To play these characters correctly today, ignore the card text and use the universal rule:
DR20: Reduce all damage by 20.
Ignore if: The attacker is adjacent AND has a lightsaber, or uses Lightsaber Throw.
Weak Spots 10: If the attacker is adjacent and doesn't have a lightsaber, the reduction is capped at 10.