The two animations shows both "mirrored" versions of Sune™. As you can see they're the same moves, just "mirrored". You need both. Note the big X. That is the target sticker of the Sune™ (explained below).
Step 6 can always be done with 2 Sunes or less. You do need to learn Sune™ to use this method.
To talk about how to use it we have to define target of a Sune™. In words, the target of a Sune™ is the corner sticker on the opposite side from the first turn, on the side of the corner moving up. But it's easier to see from the examples. Just look for where the black X's are in relation to the first move. The pattern is easy to see.
Once you understand targets, you only have these simple rules to keep in mind (assuming a red top layer):
Rule 0: With no correct corners, target one of the four red stickers. After the Sune™, there will be one correct corner. Use Rule 1 to solve it.
Rule 1: With one correct corner, target as shown in the beginner section. This solves the corners and completes step 6!
Rule 2: With two correct corners, target one of the two non red stickers of a twisted corner. After the Sune™, there will be one correct corner. Use Rule 1 to solve it.
Rule 3: Three correct corners is impossible
Rule 4: Four correct corners is already solved.
The middle sequence is a double Sune™, and the right is a Bruno™, and they're simple and fast for those corner looks.
You can get under 20 seconds with just what's on this page. Mostly by quick recognition and execution, both of which are in general more important than learning lots of sequences. Having said that, it does help a lot to learn more move sequences to avoid Step 7 more often.
There are only 49 possible positions in Step 6. Here are solutions to every single one,
mostly using combinations of Sune™ and a few other sequences.