Naming convention for sea ice thickness per unit area of ice or ocean When defining an area average of sea ice thickness, it is important to distinguish between an average taken over an ocean area

$$h_1 = \sum \frac{\text{sea ice thickness (in area)}}{\text{area}_\text{ice} + \text{area}_\text{water}}$$

and an average taken over the ice covered area

$$h_2 = \sum \frac{\text{sea ice thickness (in area)}}{\text{area}_\text{ice}}$$

The notations $h_1$ and $h_2$ are mine. I've seen both quantities referred to as “effective ice thickness”. I might go with Holland et al 2014 and call $h_1$ “effective ice thickness” and $h_2$ “average ice thickness”.

As an example think of a multi-category sea ice model: Let's say an ocean grid cell has a sea ice concentration of 50 %. Thus 50 % of the ocean grid cell is covered by water. The other 50 % of the grid cell is half-filled with 50 cm thick ice and half filled with 30 cm thick ice. For this case $h_1$ and $h_2$ would be 0.2 m and 0.4 m:

$$\begin{align}
h_1 &amp;= 0.5 \times 0 \text{m} + 0.25 \times 0.5 \text{m} + 0.25 \times 0.3 \text{m} = 0.2 \text{m} \\
h_2 &amp;= 0.5 \times 0.5 \text{m} + 0.5 \times 0.3 \text{m} = 0.4 \text{m} 
\end{align}$$ 

Both variables can be useful. As I have seen conflicting names, I am a bit puzzled with the naming for $h_1$ and $h_2$. What are the standard names for $h_1$ and $h_2$ used in the sea ice community?
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".