How can we define spin as the spin of an electron around it's own axis if an electron is described by a probability cloud of finding an electron in a point in space? How does that probability cloud spin around it's own "axis"(I find this ill-defined too) and create magnetic field? Also, when is the elctron in an atom described as a particle and follows particle principles?
Another way to look at spin, complementary to the other ways, which I find helpful is look at an abstract generalisation of the concept of angular momentum and forget about things like classical tops. This generalisation begins in something called Noether's Theorem which you probably haven't met yet. You need some background but the idea is essentially simple. If we find that a system's physics is unchanged when we impart a continuous transformation on it, then Noether's theorem tells us that there must be one conserved quantity for each such continuous transformation. Thus, Nature doesn't seem to care whether we put our co-ordinate system origin here, or there, or anywhere in between. We can slide the origin of our co-ordinate system around, but the physics stays the same. There are three continuous transformations here: sliding the co-ordinate origin in the $x$, $y$ or $z$ directions. Noether's theorem tells us that there are three corresponding conserved quantities: these are what we call the components of linear momentum. Likewise, most physical laws don't care where we put the time origin: we can slide it continuously back and forth, but the physics doesn't change. There is thus another conserved quantity: this is the one we call energy. Lastly, if we rotate our co-ordinate system around, we don't change the physics, so there are three conserved quantities, one for each of the three possible axes of rotation. These we can define as the angular momentum components.
Therefore, we could imagine being clever beings who know nothing about spinning tops, and yet we could still foretell the existence of angular momentum that is conserved for an isolated system. Electrons, photons, all kinds of particles still have this property because their physics (encoded in something called their Lagrangian) doesn't change when we impart rotational transformations on our descriptive co-ordinate system, even if we can't imagine them as spinning like a top. Our clever beings might have no sense of sight, and therefore might not get hung up on needing to see things "spin" (i.e. beget a particular, highly noticeable sight-sense experience that helped our evolutionary forebears to spot movement when hunting or fleeing predators) before they would declare those things had angular momentum. For such beings, there would be no everyday concept of spin such as we have: for them there would only be conserved quantities whose existence was inferred from applying Noether's theorem to the invariance of physics with respect to co-ordinate system rotation.
I recall being bothered by the same kind of question when I was much younger. You may be finding the assertion that something delocalised like an electron simply has angular momentum without spinning a bit of a spin out, so it may help you to know that this thought was found to be very troubling to some very great minds. Wolfgang Pauli thought the idea was preposterous precisely because he was still thinking of spinning balls, and therefore he calculated that the surface of these balls would need to be spinning at greater than lightspeed to account for known angular momentum. Quantum mechanical spin is widely cited as the first quantum property to be discovered that has no classical counterpart, hence is naturally going to seem pretty weird at first meeting.
Spin is not defined as the spin of electron around its own axis. Spin is the intrinsic angular momentum of the electron - intrinsic meaning it does not arise from the electron's motion, but is a property of electron itself.
The electron in the atom "can" be described as a particle if you are using the Bohr model of the atom. Quantum mechanical picture of the electron is a more accurate description and should be used when dealing with atom-sized objects if you can deal with the mathematics.
Electrons never follow "particle principles", by which you seem to mean the physics of classical point particles. It's only in certain cases that a classical approximation is sufficient for human purposes, i.e. when we don't care about the uncertainty relations that govern quantum mechanical objects.
In general it's better to think about elementary particles as quanta of a field. Physically measurable interactions of this field with itself are always represented by the presence (and exchange) of one or multiple of these quanta. It's the symmetries of this quantum field that give rise to conserved quantities like charge and angular momentum, including spin. It's the quantization of the field which forces quanta to carry fixed amounts of these conserved quantities.
Quantum Spin of a particle just represents another degree of freedom (e.g $+/- 1/2$ for electrons) and due to its representation as an "angular momentum operator", it is refered to as "spin". However it is not an analog (or actual revolution of a particle around its own axis). At least not in a classical sense.
In summary it represents another degree of freedom (or another dimension if you like) of a particle, which can be represented as a "(gemeralised) angular momentum".
A particle in a specific level of physical description can be taken as a (classical, point-like) particle (and not as a distributed material, like a wave) when the relatve dimensions of the overall system and the particle allow for such a description or the level of accuracy of description allows it. In general it depends on the "characteristic wavelength" (de-Broglie wavelength) of a material wave-particle and its relation to the dimensions of the overall system under study.