Naming convention for sea ice thickness per unit area of ice or ocean
It would appear that you are spot on with the commonly used terms in your question.  A well-referenced post on the Arctic forums (not by me) entitled 'Average sea ice thickness vs effective sea ice thickness', provide information about this very question.  Specifically:

Average Sea Ice Thickness = Volume of Sea Ice/Ice Covered Area

Effective Sea Ice Thickness = Volume of Sea Ice/Grid Cell Area

According to the author of the post, (an Administrator of that site), these terms are imbedded in a lot of the published literature - for example, in the article "Uncertainties in Arctic sea ice thickness and volume: new estimates and implications for trends".