Here is the trait for the new Angel from the Days of Future Past set.

If I have characters on my team using the Defenders ATA, does this qualify them to utilize Angel's trait (if adjacent)?

After all, an ATA, it is still considered a team ability and the name of the team ability is simply "Defenders", just like what is printed on the card.

I think the intent is obvious, but there are a couple examples where common words used in different places are treated as the same. (Outlaws and Atlantis keywords for instance)

No, the Defenders Additional Team Ability is not the Defenders Team Ability. He grants the use of the Defenders Team Ability.