What is the correct definition of $k$-tree?
I basically agree with you, with just a tiny modification:


  A graph $G$ is a $k$-tree if and only if either $G$ is a complete graph with $k+1$ vertices, or $G$ has a vertex $v$ such that the (open) neighborhood of $v$ forms a $k$-clique, and $G - v$ is a $k$-tree.


In other words, $v$ should have degree $k$, instead $k-1$ in your definition.

I personally prefer the bottom-up definition, but this is just a matter of taste:


  
  The complete graph on $k+1$ vertices is a $k$-tree.
  A $k$-tree $G$ with $n+1$ vertices ($n\ge k+1$) can be constructed from a $k$-tree $H$ with $n$ vertices by adding a vertex adjacent to exactly $k$ vertices that form a $k$-clique in $H$.
  No other graphs are $k$-trees.
  


This definition is a slightly modified version of the definition from Pinar Heggernes' lecture notes.