Q:

What is a hexagon? 

Having a slight parenting anxiety attack and I hate teaching my son something incorrect. 
Wiktionary tells me that a Hexagon is a polygon with $6$ sides and $6$ angles. 
Why the $6$ angle requirement?   This has me confused. 
Would the shape below be also considered a hexagon? 

A:

Yes, it would still be considered a hexagon.  The reason why we require "$6$ angles" is probably just because you don't want the $6$ lines to cross and create "more than $6$ angles".  The hexagons you are probably thinking of (the one with all the angles and sides equal) would be regular hexagons. 
Hope that helps. 

A:

Yes.  It also has six angles, but one of them is greater than $180^{\circ}$. 

A:

A  hexagon is a simple closed polygon $P$ (presumably in the plane) with $6$ sides.  Simple means that the sides are arranged in cyclic order, that two subsequent sides have exactly one point, called a vertex, in common, and that otherwise the sides don't intersect.  Maybe you want to include the condition that subsequent sides should not be parallel.   Jordan's curve theorem then guarantees that $P$ has a well defined interior (an "open hexagon") and an exterior which extends to infinity. 
The polygon in your figure is a hexagon according to this definition.  If you want to exclude it you would have to  explicitly require that the considered hexagon be convex. ( Maybe you can use this opportunity to make your son familiar with the notion of convexity $\ldots$)

