Isn't linear time O(n)?
Usually we call statement $A$ stronger than $B$ when $A$ implies $B$: $A \Rightarrow B$ (weaker-stronger). In other words, $B$ is weaker than $A$.
When the presenter is speaking about linear time for partition, this is a stronger statement than $O(n)$ time. All linear functions are in $O(n)$, but it also contains non-linear functions.
For example: $\sin n, \frac{1}{n}, \sqrt{n} $ are all in $O(n)$, but they are not linear. As was written in a comment, $O(n)$ is the set of functions bounded by linear functions, but not the set of only linear functions.
To be linear gives more information than to be in $O(n)$, to be in $\Omega(n)$, even to be in $\Theta(n)$.