The Water Bucket (Full) is an item utilized during the Enlightenment quest, which is obtained from Bando Buno of Cloudopolis.
Bando will give the player a Water Bucket, which can then be filled at the nearby water pump in order to create a Water Bucket (Full), which can then be used to clean the dojo.
When the Water Bucket (Full) is used at one of the quest markers to clean the dojo, it'll empty its contents, reverting back into a regular Water Bucket.