Sensitivity on differential privacy
The differential privacy book is the typical reference for the area, and it is quite useful here. Since this answer essentially amounts to quoting from that book, I'll walk through how to find the right things to quote.
Ctrl+F-ing &quot;Laplace&quot;, we find Theorem 3.6, which states that the Laplace mechanism is $(\epsilon,0)$-differentially private.
This mechanism adds i.i.d. $\mathsf{Lap}(\Delta f/\epsilon)$ noise to the output, where (as you mention):
$$\Delta f = \max_{\substack{x, y\in\mathbb{N}^{|\mathcal{X}|}\\\lVert x - y\rVert_1 = 1}} \lVert f(x) - f(y)\rVert_1$$
So this is the $\ell_1$ version of sensitivity.
Ctrl+F-ing &quot;Gaussian&quot;, we see that it works for sensitivity defined via:
$$\Delta_2 f = \max_{\substack{x, y\in\mathbb{N}^{|\mathcal{X}|}\\\lVert x - y\rVert_1 = 1}} \lVert f(x) - f(y)\rVert_2$$
This is an $\ell_2$ notion of sensitivity (although note that &quot;neighboring datasets&quot; $x, y$ are still within 1 of eachother in the $\ell_1$ norm, meaning they still differ in at most one row).
Theorem 3.22 then shows that to be $(\epsilon, \delta)$ differentially private, the Gaussian mechanism adds i.i.d. noise $\mathcal{N}(0, 2\ln(1.25/\delta) \Delta_2(f)^2/\epsilon^2)$ to the output of the function.