Bone flip always uses the world axes. I don't think any other software does this, because it isn't described by a transform matrix. It does everything as in your first GIF, then flips the position of the blue box over the world Y axis, using the red box world position as the origin for the flip.

It's weird. It's changing. Scale -1 will do the same thing as flip, as pretty much everyone expects it to. We've considered removing flip, since it just uses -1 scale under the covers. However, we may keep it because there's a difference in knowing a character is flipped versus it's scale is exactly -1. Usually this comes up when integrating with a physics engine. Maybe flip can be kept as a skeleton property, like it was long ago, which just flips the entire skeleton over the world origin. For anything else, people can use -1 scale.

Additional consideration for the separation:
Skeleton flip may be used for persistent "facing state" of a character, to reflect whether the character's logic is making it face left or right.
A separate animated root bone flip may be used for transient, nonlogical flipping as in the cases of:

• animating a turn animation (to include the character facing the opposite direction before facing the final one) or...
• animating a spinning animation while allowing it to end facing the correct direction.

In case you're removing skeleton.flip, I guess the user can just have the root bone (for logic) and an intervening bone (for transient flips) before the rest of the hierarchy.