Definition of prefixes of a string
No.  As you have defined it, $t$ is only in the language if it is every prefix, except $\varepsilon$, of $s$.  That definition describes $\emptyset$ if $|s|$ is not 1, or $\{s\}$ if $s$ is a single symbol.  If you change ∀n∈{1,…, length(s)},  to $\text{s.t.}$ $1 \leq n \leq |s|$, then it would be almost equivalent.  However the first language includes the empty string.  

Also "first n part of" is not a very good way to describe a prefix of length n.  It could be ambiguous.  And you should define what $t$ is, i.e. $t \in \Sigma^*$.