Classical and quantum probabilities in density matrices
Yes, the density matrix reconciles all quantum aspects of the probabilities with the classical aspect of the probabilities so that these two "parts" can no longer be separated in any invariant way.
As the OP states in the discussion, the same density matrix may be prepared in numerous ways. One of them may look more "classical" – e.g. the method following the simple diagonalization from equation 1 – and another one may look more quantum, depending on states that are not orthogonal and/or that interfere with each other – like equations 2.
But all predictions may be written in terms of the density matrix. For example, the probability that we will observe the property given by the projection operator $P_B$ is $$ {\rm Prob}_B = {\rm Tr}(\rho P_B) $$ So whatever procedure produced $P_B$ will always yield the same probabilities for anything.
Unlike other users, I do think that this observation by the OP has a nontrivial content, at least at the philosophical level. In a sense, it implies that the density matrix with its probabilistic interpretation should be interpreted exactly in the same way as the phase space distribution function in statistical physics – and the "quantum portion" of the probabilities inevitably arise out of this generalization because the matrices don't commute with each other.
Another way to phrase the same interpretation: In classical physics, everyone agrees that we may have an incomplete knowledge about a physical system and use the phase space probability distribution to quantify that. Now, if we also agree that probabilities of different, mutually excluding states (eigenstates of the density matrix) may be calculated as eigenvalues of the density matrix, and if we assume that there is a smooth formula for probabilities of some properties, then it also follows that even pure states – whose density matrices have eigenvalues $1,0,0,0,\dots$ – must imply probabilistic predictions for most quantities. Except for observables' or matrices' nonzero commutator, the interference-related quantum probabilities are no different and no "weirder" than the classical probabilities related to the incomplete knowledge.
Let's look at a famous, concrete example: Perfectly unpolarized light.
Alice creates unpolarized light by randomly (incoherently) mixing left-circular-polarized light with an equal intensity of right-circular-polarized light.
Bob creates unpolarized light by randomly (incoherently) mixing vertically-polarized light with an equal intensity of horizontally-polarized light.
There is no measurement that will tell you which light is Alice's and which is Bob's.
Are Alice's light fundamentally the same as Bob's light, or are they different kinds of light that are impossible to tell apart?
Well, one shouldn't make too much of these kinds of questions. But if I had to choose, I would say that they are different kinds of light, because the classical incoherent mixing process leaves a trail of information out there that is sufficient to tell the two beams apart (even though I may not have that information right now in practice).
For example, maybe Alice and Bob are each combining two different laser beams with slightly-different (and randomly fluctuating) frequencies. (This is a legitimate way to incoherently add two light beams in practice.) If I don't have a very fancy spectrometer, I can describe all my possible measurements by saying that these are unpolarized beams. But if I do have a fast and high-resolution spectrometer, I can figure out which beam is Alice's and which is Bob's.
This is an example of a broader truth: Classical probabilities are more situation-dependent than quantum probabilities. Specifically: If two people each think that a particle is in a pure state, they will always agree on what state it is in, and therefore they will agree on the probability distribution for any possible measurement of that particle. But if two people each think that a particle is in a mixed state, then they will often disagree on what mixed state it is in, because they may have different auxiliary knowledge, which leads them to assign different classical probabilities. (For example, maybe the particle is one of an EPR pair, and its twin has been measured, but only one of the observers knows the measurement result.)
But, given a state of "my knowledge right now", there is no way to draw a line between classical probabilities and quantum probabilities---and no reason to!
I will provide an answer but from a different perspective, and hopefully convince you that there is information in a density matrix which has no classical counterpart. Furthermore this can hence be considered a quantum component, and it can be shown that this information is stored as the eigenvectors of $\rho$.
I will give an example of how this manifests. The Fisher Information $I(\theta)$ is a statistic from classical probability theory which characterises how quickly one can learn about a parameter $\theta$ which characterises a probability distribution $p(\theta)$.
Specifically the variance of an unbiased classical estimator $\hat{\theta}$ respects the Cramer Rao bound $$\mathrm{var}(\hat{\theta})\geq \frac{1}{I(\theta)}$$
The additivity of information means that if you sample the distribution $n$ times, collecting measurements each time the expected error $\Delta \theta_c = \sqrt{\mathrm{var}(\hat{\theta})}$ of any estimator goes like $$\Delta \theta_c \propto \frac1{\sqrt{n}}$$
This is recognised in the scaling of the standard deviation $\sigma$ in things like central limit theorem.
We can define a quantum analogue, to the fisher information $J(\theta)$ which satisfies an analogus bound, known as the Quantum Cramer Rao bound.
However it is found that by permitting entanglement between classically independent sampling events, the bound is much better. And after having collected a dataset of $n$ measurments, the best possible quantum estimator is bound only by the error $$\Delta \theta_q \propto \frac1{n}$$.
This shows that a general quantum state $\rho$ can definately support statistics which a classical probability distribution cannot.
The quantum Fisher information of a density matrix which depends on a parameter $\theta$ $$\rho(\theta) = \sum_i p_i(\theta) |\psi_i(\theta)\rangle\langle\psi_i(\theta)|$$ can be seen to seperate into several contributions, one of which is the classical Fisher information of the spectrum $p_i(\theta)$, another of which is a Fubini-Study like term which accounts for the information stored in the basis $|\psi_i(\theta)\rangle$. The possibility of (super-classical) quantum scaling depends entirely on the existence of this quantum term.
Alternatively stated, in terms of the behaviour of the Fisher information statistic and its quantum analogues, a density matrix $\rho$ supports non classical behaviour only if the basis set $|\psi_i(\theta)\rangle$ contains information relevant to the measurment, and in this sense, information stored in this way may be considered non-classical.
Useful stuff
If you are interested in some of the topics discussed here see this good review for an explanation. http://arxiv.org/pdf/1102.2318v1.pdf
This for an accessible but mathematical explanation of the QFI. http://arxiv.org/pdf/0804.2981.pdf