In this question, I mentioned that the tag avatar-the-last-airbender may not the best choice for the question, as the tag only covers the first series, Avatar: The last Airbender. The question, however covers both, Avatar: The last Airbender and The Legend of Korra. Should there be a second tag the-legend-of-korra, or should the Avatar-Tag be generalized? A new tag can make searches more specific, however both series are simply referred to as Avatar, most times.
I think that in this case, the avatar tag should be created and used for both The Last Airbender and The Legend of Korra. Madara's example of having a naruto and naruto-shippuuden tag to tell the two series apart isn't a very good one since everybody just uses the plain Naruto tag for both series.
To deal with differentiating the two series, I propose that avatar be used for both with related synonyms for avatar-the-last-airbender and legend-of-korra. This way searching "avatar" will yield results for both series as they're both commonly referred to as Avatar and searching for either of the synonyms will give results for those particular series.