Why is magnetism used to refer to two seemingly distinct phenomena? I've seen two explanations of magnetism that seem to be describing two completely different things.

$(1)$ Magnetism is caused by electric spin, denoted $m_s$, equalling $\pm \frac{n\in \Bbb N}{2}$. A particle with spin is magnetic, the direction of its spin determining it's north and south pole.  For not-so-big atoms, electron  configurations (usually?) follow Hund's rule, meaning that the slots in an orbital are filled up with electrons all of the same spin (so as to minimize same-charge repulsion). In atoms with half-filled orbitals, this renders the atom a magnet due to the uncancelled spins of electrons giving it a net magnetization. If the atoms, at the crystal scale, order themselves with aligned spins (ferromagnetically), then the grain becomes magnetic. If all the different grains, and at the next level, phases, align themselves magnetically, then the entire mass becomes magnetic.

$(2)$ Magnetism is a force felt by a charged observer outside of the reference frame of moving charges. The observer will feel an (additional) force imposed by the moving charges, other than their normal, Coulombic electrostatic force. This is due to length contraction caused by the charges' movement making the observer experience the  charges as more densely packed from an external reference. A higher density of charges means a higher electrostatic force. This is why a conducting wire, that quantity-wise should be neutral, winds up with a magnetic field; there's an equal amount of electrons and protons, BUT, the electrons are moving and thus contracted, making any slice of the wire more dense with electrons than protons, from an external frame of reference.

This Veratasium video, and this MinutePhysics follow-up, explain magnetism as being the name for both of these forces. However, I don't see how these forces are at all the same. One is just electric spin $(1)$, whatever that is, and the other is just (vaguely put) an extra electrostatic force caused by length contraction $(2)$. I'd like an explanation for how $(1)$ and $(2)$ are actually the same.
 A: Magnetic field as a relativistic effect
Unfortunately, the Veritasium videos contain some truth but follow a misleading teaching tradition, going back to Purcell's book on Electromagnetism, which presents the magnetic field as a relativistic effect.
There are different reasons this claim is false.

*

*Special Relativity shows that there is a unique tensor quantity, the electromagnetic field, whose components are both the $E$ and the $B$ fields. Moreover, $E^2-B^2$ and ${\bf E}\cdot {\bf B}$ are relativistic invariants, easily obtained from the tensor field. Therefore, if ${\bf E}\cdot {\bf B}=0$, and $E^2-B^2$ is positive, it is possible to find an inertial reference frame where the $B$ field is zero, while if it is negative, this is never possible (but it is possible to find a reference frame where the electric field is zero).

*As Jefimenko showed more than 25 years ago, if one can think of the magnetic field as a relativistic effect, it would also be possible to think of the electric field as a relativistic effect. The two possibilities are not consistent with each other. Again, this fact points to the full electromagnetic field as a unique quantity made by two independent components, the electric and the magnetic fields.

*Maxwell equations in a single reference system show that both the electric and magnetic fields are necessary to describe a scenario where both charge density and current are present.

Purcell's approach shows that the relativistic consistency of the description of the physical effects requires properly taking into account the relativistic transformations of the sources (electric density and currents) and the fields (electric and magnetic).
To summarize, let's forget about the magnetic field as a relativistic effect. In a given reference frame, electric and magnetic fields are required to describe the effects of charges and currents. In the particular case of stationary sources, one can separate the electric field due to the charge density and the magnetic field due to the current density.
Magnetic moment due to the spin
This is not something really different from the usual magnetic fields due to currents. It has been shown that the quantum description of particles with spin implies a probability current density due to the spin  (see Mita, K. (2000) Virtual probability current associated with the spin. American Journal of Physics, 68(3), 259-264). If the particle is charged, this probability current density is a current density that acts as a source of a (spin) magnetic field. Starting from the spin magnetic moment, quantum mechanics explains the ferromagnetic or antiferromagnetic couplings as a phenomenon connected to the electrostatic advantage of a parallel or antiparallel spin ground state.
Starting from this essential step, one can understand the ferromagnetism of macroscopic samples in terms of aligned domains.
In summarizing, there are no competing explanations for magnetism. The basis is electromagnetism as encoded in Maxwell's equations. Electric currents are always the basis of static magnetic fields. Spin is a key ingredient for two reasons: i) it implies the presence of microscopic magnetic moments, and ii) it requires the antisymmetry of the wavefunctions at the basis of the possibility of ferromagnetic couplings. Notice that the magnetic moment due to the spin can also be interpreted in terms of a current density connected to the multicomponent nature of the wavefunctions.
A: I would say there is no good answer to this because what you are asking for is an explanation of where the spin magnetic moment of elementary particles comes from. Ask Derek: https://www.youtube.com/watch?v=hFAOXdXZ5TM&t=106s
Magnetism originating from spin magnetic moments or from electric currents are the same in that both phenomena can be described using Maxwell's equations (albeit with different sources) to yield the same kind of magnetic fields that can be experimentally observed through their common effect on permanent magnets like the compass needle in the video or their Lorentz force on moving charges.
