First of all, it should be intuitively clear that if you combine two noise models with similar "strength" that this results in a "noisier" model. The noise model should fit to your experimental data, e.g. the fidelity that you or your collaborators have measured.
IMO, the first fidelity has nothing to do with the second. Given a unitary $U$, the channel that you describe has the form
$$
\mathcal E (\rho) = p U\rho U^\dagger + (1-p)\rho.
$$
Let us denote the channel defined by $U$ as $\mathcal U$. Using some standard identities, the (average gate) fidelity of $\mathcal E$ with $U$ is
$$
F_\mathrm{avg}(\mathcal E,\mathcal U) = p F_\mathrm{avg}(\mathcal U,\mathcal U) + (1-p) F_\mathrm{avg}(\mathrm{id},\mathcal U)
= p + (1-p) \frac{|\mathrm{tr} U|^2 - d}{d(d+1)}.
$$
Note that this is equal to $p$ if and only if $U$ is (proportional to) the identity matrix. Of course, you may change the definition of $p$ accordingly.
The reason why the fidelity is not just the "success probability" in the above sense, is that any unitary is acting trivially on its eigenvectors,
$$
U |\psi\rangle\langle\psi| U^\dagger = e^{i\varphi}|\psi\rangle\langle\psi|e^{-i\varphi} = |\psi\rangle\langle\psi|,
$$
hence, it acts as the identity on at least a $d$-dimensional subspace of operator space.
As a final note: Given the fidelity of a gate, you do not know anyhing about the type of noise in your system. Ideally, you would come up with a noise model that is motivated by the physics of the platform that you are using. I'm not sure what the motivation behind the channel $\mathcal E$ is. To me, it looks more like some kind of "detector noise", i.e. either the detector clicks, or it doesn't. For gates, this seems weird. In the end, you are hitting your system with a laser, RF pulse, whatever, so you're definitely doing something in any case. If you do not know anything about the system, you should rather use depolarizing noise.
