The definition of locality seems to be arbitrary, am I correct?

There is at least one different point of view on the question. More frequently, you say that a property $\mathcal P$ holds locally at a point $x$ of a topological space $X$ if for every neighbourhood $U$ of $x$ there is a neighbourhood $V$ of $x$ such that $V\subset U$ and $V$ enjoys $\mathcal P$. In other words, if there is a neighbourood basis $\mathcal U$ at $x$ such that every element of $\mathcal U$ enjoys $\mathcal P$. This is the case for local connectivity, local path-connectivity and many others.

In some, less frequent, cases (for instance, local compactness) this in fact has the other meaning, that is, there is a neighbourhood of $x$ enjoying $\mathcal P$. But that's definitely rarer.

From the commenter above Don Thousand: "When we say something is true locally, it means that there exists a neighborhood in which it is true. That neighborhood can be different for each point."