There are indeed other such polygons.
-- For example there is one for $n = 11$, as follows
(the origin is in the lower left corner):

Also there is one for $n = 15$:

Further there are $21$ such polygons for $n = 16$.
One of them is the following:

These polygons have been found with this GAP function.

There is one such polygon for $n = 8$, $n = 11$ and $n = 15$, each,
there are $21$ such polygons for $n = 16$, and $225$ for $n = 19$.
For all other $n < 20$ there are no such polygons.

The complete list for $n \leq 19$ in GAP-readable form
can be found here.

**Added on May 3, 2016:** A zip file with *.png images of all
$249$ polygons for $n \leq 19$ can be found here (545KB).

**Added on May 4, 2016 (following a request in the comments):**
When leaving away the condition that the polygon is entirely in the
first quadrant, we get $3$, $5$, $6$, $584$ and $882$ distinct polygons
for $n = 8$, $n = 11$, $n = 12$, $n = 15$ and $n = 16$, respectively.
A GAP-readable coordinate list can be found here, and
a zip file with *.png images of all these $1480$ polygons
can be found here (2.8MB).

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