Is it correct to say a quantum particle is in both "states" at the same time? In popular media, and even in introductory books it is common to say that quantum objects are characterized by the non-intuitive notion of being in two or more eigenstates at "the same time".
An example about quantum computing (http://www.newsweek.com/quantum-computing-ibm-580751)

While traditional computers put bits in 0 and 1 configurations to calculate steps, a qubit can be a 0 and a 1 at the same time.

Isn't this definition misleading to the public in general? For me, superposition is not about two or more eigenstates coexisting at the same time, it is just a physical phenomenon that is mathematically convenient when we are dealing with probabilistic systems.
I am aware there is a lot of controversy even in the philosophical interpretation of quantum mechanics itself, which is not my focus to discuss the various interpretations here.
So, based on the grounds of quantum mechanics, is it correct to say that "eigenstates" "coexist" at the same time? Is the media and the common sense incorrect or is it just a convenient naïve approximation?
 A: Yes, it is wrong to say they coexist. It is better to say that the quantum state is in a combination of both of the eigenstates. I think it conveys a similar meaning to a layperson whilst also being more technically correct.
Quantum states are linear combinations of the eigenstates of the observable. They're vectors, and they exist in a vector space and do vector things like adding and dot-producting (actually they exist in an inner product space to make the braket mean something but that's by-the-by).
You would never say that a 2D vector like $\vec{r} = \frac{1}{\sqrt{2}}\left(\mathbf{x} + \mathbf{y}\right)$ is pointing in the '$\mathbf{x}$' direction and the '$\mathbf{y}$' at the same time. You would say that the vector is a combination of the '$\mathbf{x}$' and '$\mathbf{y}$' directions. Depending on how you look at the vector, you might project it onto $x$-axis or the $y$-axis. This corresponds precisely to projection operators in quantum mechanics (things like $|0\rangle\langle0|$).
A: In my opinion it is wrong, and the semantic confusion started with Schrodinger's cat being alive and dead at the same time.
The confusion with Schrodinger's cat comes from making a fancy detector out of a live animal. What one is measuring with opening the box and finding a dead cat  is a single measurement point in the quantum mechanical  probability of the nuclide to decay, a   calculable state. A throw of the quantum mechanical dice. One would need many boxes to measure the distribution, and many dead cats.
All we know about the nuclide is that it is in an excited state and it has a probability of decay.
In the example in question, the qubit has a probability of being either zero or one, given by quantum mechanical calculations. Only an interaction/measurement at a given time can decide which state it is in.  In my cat example the interaction is the nuclear decay (starting the poison).
A: 1)  If there is anything (a qubit or anything else) that is equal to both $0$ and $1$ at the same time, then $0=1$.  I think we can say this is unambiguously flat-out wrong, and nothing more needs to be said.  I will say a little more anyway.
2)  The state of a particle by definition contains everything there is to know about that particle.  This makes it impossible by definition for a particle to occupy more than one state at a time.
3)  Every state of every quantum system is (in multiple ways!) a superposition (i.e. a sum) of other states in exactly the same sense that the integer $8$ is a superposition of $3$ and $5$.  The number of planets is $8$.  Most of us don't want to express this by saying that the number of planets is both $3$ and $5$ simultaneously.  You can of course go around saying such things if you want, but nobody is going to understand you.  
A: There are few if any unequivocal statements that can be made about interpretations of QM, nearly everything is just someone's opinion. No one really understands what superposed quantum states actually are. In my opinion, quantum states always manifest as multiple states at once as a result of any single measurement, because single quantum states are ideal pure states that would have zero entropy, and zero entropy is not realizable by the Nernst statement of the third law of thermodynamics. All realizable lab measurements are spectrally limited in any measurement basis and thus must necessarily result in quantum mixed states, i.e., states with nonzero entropy, they are incoherent superpositions. This means that quantum states (often considered mutually exclusive) actually must exist concurrently.
A: 
Isn't this definition misleading to the public in general? For me, superposition is not about two or more eigenstates coexisting at the same time, it is just a physical phenomenon that is mathematically convenient when we are dealing with probabilistic systems.

Quantum mechanics isn't primarily about probabilistic systems. The square emplitudes of states sometimes obey the rules of probability, but often break those rules:
https://arxiv.org/abs/math/9911150.

I am aware there is a lot of controversy even in the philosophical interpretation of quantum mechanics itself, which is not my focus to discuss the various interpretations here.

You have badly misunderstood the situation concerning the 'interpretations' of quantum mechanics, as physicists usually do. Accounts of quantum mechanics that say a particle is in two states at once are closer to the mark than your current ideas.
Different 'interpretations' are, at best, different accounts of what is happening in reality. Many of those accounts are not about quantum mechanics at all. Rather, they are about entirely different physical theories that make different predictions, e.g. - the predictions of the pilot wave theory don't match those of quantum mechanics:
https://arxiv.org/abs/1510.03508
A theory that gives a different account of how the world works from quantum mechanics and makes different predictions can hardly be described as an interpretation: it is a distinct physical theory. I don't agree with pilot wave theory, but it has potential to be a viable competitor. The GRW theory and other theories a physical account of collapse are in the same boat.
The Copenhagen and statistical interpretations are just bad philosophy dressed up as physics. They deny or obscure the need for an explanation of what is happening in reality. This is a philosophical issue, but like many such issues it is also a matter of immediate practical importance. People think philosophy is some pie in the sky nonsense, but that's only true of bad philosophy, like the philosophy underlying the Copenhagen and statistical interpretations. Without an account of what is happening in reality, it is impossible to set up or understand experiments properly. For an example, see the unnecessary confusion over whether or not quantum mechanics is local:
https://arxiv.org/abs/1109.6223
https://arxiv.org/abs/quant-ph/9906007
There is only one interpretation of quantum mechanics itself, and it involves multiple interacting versions of the same system, sorted into structures that approximately resemble the universe as described by classical physics:
https://arxiv.org/abs/quant-ph/0104033.
Those different versions interact, which makes the world more complicated than a non-interacting collection of parallel universes. However, the correct interpretation does involve multiple versions of the same system that may be in different places, so the popular article is a lot closer to the mark than your ideas, which are just recycled instrumentalism.
