Is there any difference in meaning between these two statements I'm reading text that states

the vector $\ x \in \mathbb{R}^{n} $ is $\ s$-sparse if at most $\ s<n$ of its entries are non-zero

Is that any different from stating:

the vector $\ x$ is $\ s$-sparse if $\ s<n$ of its entries are non-zero.

I don't see the difference and I find the first redundant; if I'm stating that $\ x$ is $\ s$-sparse then could $\ x$ have anything other than $\ s$ non-zero entries?
Definition of an $\ s$-sparse:
A vector $\ x \in \mathbb{R}^{N} $ is called $\ s$-sparse if at most $\ s $ of its entries are nonzero, i.e. if
$\ \|x\|_{\ell_0} := card(supp(x))\le s $ 
where $\ supp(x) := \{j \in [N] : x_j \ne 0\} $

The vector $(1, 0, \text{<500 more zeros>})$ is 12-sparse according to the first definition, but not according to the second.  
