A compound Poisson process with jump intensity measure (Lévy measure) $\nu$ is a Lévy process $X_t$ on $\mathbb{R}^d$ such that $X_{t+s}-X_s$ has law $\pi_{t\nu}$, where $\pi_{t\nu}$ is the measure with characteristic function $p \mapsto\exp(-t\int (1-e^{ip\cdot y} \, \nu(dy))$. (Definition taken from the lecture notes I'm currently looking at).
That definition suggests that the intensity measure $\nu$ indicates the size of the jumps when they occur. However, in other contexts, it's seemingly also used to denote the measure governing how often jumps occur, which in the above example seems to be just the Lebesgue measure.
How can one separate the terminology here? "Intensity measure" seems to be quite an ambiguous concept.