It is known, that the index of a Lie algebra is a semi-invariant for degenerations
(by Ooms and Elashvili), i.e., if $L_1$ degenerates to $L_2$, then
$ind(L_1)\le ind(L_2)$. This is very useful.
For example, it follows that any filiform Lie algebra of dimension $n$ has index
less or equal than $n-2$, where only the standard graded filiform $L(n)$, which you have
defined above, has exactly index $n-2$.
In general, there are many other Lie algebras of dimension $n$ and index $n-2$, e.g.,
also the quasi-filiform Lie algebras $L(n-1)\oplus \mathbb{C}$. See here also the work
Adini and Makhlouf. The Hasse-diagram of complex nilpotent Lie algebras in dimension 6
gives explicit examples, e.g., we have degenerations from the top algebra $L_{6,20}$
as follows (notation of Magnin for the Lie algebras)
$L_{6,20}\rightarrow L_{6,18}\rightarrow L_{6,17} \rightarrow L_{6,16} \rightarrow
L_{5,5} \oplus \mathbb{C}\rightarrow \mathbb{C}^6$, with index numbers
$2 \rightarrow 2 \rightarrow 2 \rightarrow 4 \rightarrow 4 \rightarrow 6$. See my paper
arXiv:0911.2995 for this, and a discussion on the maximal dimension of an abelian
subalgebra $\alpha (L)$, which is related to the index by $\alpha (L)\le (\dim (L)+ind (L))/2$.