Definition:Path (Graph Theory)

A path is a trail in which all vertices (except perhaps the first and last ones) are distinct.

If the first and last vertices are the same, a path is called a cycle.

Comment
It is clear that a path can also be defined as a walk in which all vertices are distinct.

By this definition it appears at first glance that a path is automatically a trail, because if an edge were to be retraced in any walk, then so would the vertices at either end of it.

However, this definition would then allow $$A \to B \to A$$, for example to be referred to as a path, and therefore a cycle, and this is not intended.

Differences in Terminology
Some sources call this a simple path, and use the term "path" to define what we have here as a walk.