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 the vertices at either end of it would necessarily be visited more than once.

However, under this looser definition, the walk $$A \to B \to A$$, for example, would fit the definition of a path, and therefore be 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.