Editorial board membership is understood and managed differently between journals and may extend from those who review often for a journal up to the Editor In Chief.  They are generally publicly recognised on a journal's website.  If you feature in such a list, you should feel free to add that Editorial Board membership to your Publons profile.

Handling editor records on Publons are intended to represent the work done in managing the peer-review process for a particular manuscript and making a final decision on the journal's behalf.