They said goodbye at a given position and are then leaving each in a different direction. They start to move at the same time from the same point.

(I don't even speak English every day anymore, so I could be wrong).

You're not wrong. I think some of it is the difference between casual speech and formal writing (people are more likely to say "after being" but write "after having been", especially in published work)**, but some of it is also just poorly phrased. It makes enough sense to a native speaker to get what the problem is asking, though.

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** I think the first may be correct in some cases, but idk the rule.