When we say an algorithm runs in $O(\lg k)$ time, that is an asymptotic statement. It means that there exists a constant $c$ such that when $k$ is sufficiently large, the running time is $\le c \lg k$. It says nothing about what happens when $k$ is small.

In particular, if we say an algorithm runs in $O(\lg k)$ time, that doesn't necessarily mean that when $k=1$ the running time is 0.

The same kind of comment applies to your question as well.

When we say an algorithm runs in $O(\lg k)$ time, that is an asymptotic statement. It means that there exists a constant $c$ such that when $k$ is sufficiently large, the running time is $\le c \lg k$. It says nothing about what happens when $k$ is small.

In particular, if we say an algorithm runs in $O(\lg k)$ time, that doesn't necessarily mean that when $k=1$ the running time is 0.

The same kind of comment applies to your question as well. No, it doesn't mean that the running time to sort an array where each element is at most one place away from its sorted position is $O(0)$. Rather, the complexity of that task is $O(n)$.