Is there a name/interest for regular languages that have a non-ambiguous ending? The basic idea is to have one or more symbol that clearly indicate the end. For example:

Non-ambiguous:


  $ab^*c$
  $(a|b)c$
   $ab^+c$
   $ab?c$
   $a(b|c)$
   $c(ab)^*ccc$
   $acc^*d$
   $abc|bcd$  


Ambiguous:


  $abc^*$
  $abc^+$
  $abc?$
  $acc^*$ or $ac^*c$


An alternative definition of a non-ambiguous ending would be that the corresponding DFA can have multiple final states, but none of them can have a outgoing transition.
According to your alternative definition, you're looking for a  language such that $u\in L\Rightarrow \forall v\neq\epsilon, uv\notin L$ (where $\epsilon$ denotes the empty string).

That property defines a useful kind of language in coding, called prefix-free codes. Note that this language class doesn't include nor is included in regular languages.

So what you're looking for is the intersection of those two classes. That would be a prefix-free regular language.

As for the interest, it seems to have some, since a google search returned this research paper.