I can give some indication of what is going on here, but I have not studied all the references as yet. This topic is complicated by the fact that there are several (possibly independent) ingredients to this issue. You can follow up on some or all aspects.
General Relativity and Einstein-Cartan Theory
The basic equations of GR we can write as $G^{ab}=P^{ab}$. Here the LHS is the geometric content of the theory and the RHS is the matter (stress-energy tensor) content of a given solution. Basic properties show that both Tensors (and several others) are symmetric. The geometric solutions are described by a metric which generates curvature. Cartan discovered later that this was not the most general form of such a geometry. Essentially the geometry could contain non-symmetric Tensors too, say $T_{a}^{bc}$ as the Torsion tensor. Thus a manifold could have two independent types of non-Euclideanness: Curvature and Torsion. The curvature in GR is well known, but the Torsion is set to zero.
It is possible to have a manifold which has zero curvature (so all parallel lines stay parallel), but has non-zero Torsion. So how does this manifest itself in the manifold? Well basically given say a unit vector (eg pencil with a point) moving it along a curve will result in the pencil having rotated from its original direction. Parallel lines stay parallel, but the space has a sort of built-in helix structure causing an intrinsic - or geometric - rotation.
Interesting mathematical idea, but does it relate to anything physical? Einstein became excited by the idea for a time, developing Teleparallel gravity - lots of torsion but no curvature (hence the Parallel aspect).
This theory was developed further in the 1960s by coupling (ie equating) the Torsion to the "Spin Tensor" $S_{a}^{bc}$. This added a second equation to the usual Einstein Tensor equation, generating EC (KS) Theory. The spin tensor is classically defined as a generalisation of angular momentum to tensors. So in the EC theory the angular momentum of an object also has gravitational effect (calculations show that it would only be significant in high spin neutron stars or maybe early universe.)
Spin Density
A simple-seeming generalisation of the idea of density from mass density ie $\rho(x)$ to linear momentum density to spin density $s(x)$ is associated with the name of Wessenhoff in 1947 who was trying to develop a "classical" Dirac theory. Here a spin value is associated with each point in space - at least in the mathematics. Later there is discussion of a "Wessenhoff fluid" - which is a strange quasi-classical fluid with vortices at all length scales. Since then the phrase "spin density" has become used in some parts of the literature without explicitly including quantum effects, but corresponds to a use of this mathematical idea.
Some authors seem to map the idea directly to quantum spin without considering such quasi-classical models - so the ideas can do double-duty.
Geometric Algebra
David Hestenes has expressed all the above concepts in Geometric Algebra (known as STA) which also allows discussion of spin density in this sense.
A paper on Spinning (Wessenhoff-like) fluids in Cosmology (under EC theory): http://deepblue.lib.umich.edu/bitstream/2027.42/49195/2/cq940917.pdf
A paper by David Hestenes:
http://geocalc.clas.asu.edu/pdf-preAdobe8/Decouple.pdf
Intrinsic Angular Momentum
Also one should make a more elementary comment about the possible use of this phrase in Classical Mechanics. A given system of particles will have an angular momentum M' in a given frame K' in which it is otherwise at rest. From the perspective of a different frame K there will be a translation formula:
M = M' + R x P
where R is the radius to centre of mass in K and P is the momentum in K. Thus from the perspective of K the angular momentum consists of the "intrinsic angular momentum" M' plus the angular momentum of the whole system. [Landau/Lifshitz Vol1 (9.6) actually use the expression "intrinsic angular momentum".]
That spin follows the angular momentum algebra is no accident - like angular momentum, it is part of the conserved quantity - the Noether charge - associated to rotations.
The reason why the $\mathfrak{so}(3)$ transformations of spin should be indeed those associated to the $\mathfrak{so}(3)$ of spatial rotations is not answerable in QM alone - you have to take it "on faith" or rather, as an experimental fact that spin is indeed (part of) the Noether charge associated to spatial rotations and not some other $\mathfrak{so}(3)$. But, when you enter QFT, you will find that every quantum field should transform in some representation of the spatial rotation group (or rather in relativistic QFT, in some representation of the Lorentz group, of which the rotations are a subgroup), and that is exactly what spin then is - the "label" of the representation the quantum field transforms in.
Since orbital angular momentum is what comes from the quantization of classical mechanics as the Noether charge of the spatial rotations, you then find that your total quantum Noether charge for the rotations will have become the sum of spin and angular momentum.
Best Answer
Note As David pointed out, it's better to distinguish between generic angular momentum and orbital angular momentum. The first concept is more general and includes spin while the second one is (as the name suggests) just about orbiting. There is also the concept of total angular momentum which is the quantity that is really conserved in systems with rotational symmetry. But in the absence of spin it coincides with orbital angular momentum. This is the situation I analyze in the first paragraph.
Angular momentum is fundamental. Why? Noether's theorem tells us that the symmetry of the system (in this case space-time) leads to the conservation of some quantity (momentum for translation, orbital angular momentum for rotation). Now, as it happens, Euclidean space is both translation and rotation invariant in compatible manner, so these concepts are related and it can appear that you can derive one from the other. But there might exist space-time that is translation but not rotation invariant and vice versa. In such a space-time you wouldn't get a relation between orbital angular momentum and momentum.
Now, to address the spin. Again, it is a result of some symmetry. But in this case the symmetry arises because of Wigner's correspondence between particles and irreducible representations of the Poincaré group which is the symmetry group of the Minkowski space-time. This correspondence tells us that massive particles are classified by their mass and spin. But spin is not orbital angular momentum! The spin corresponds to group $Spin(3) \cong SU(2)$ which is a double cover of $SO(3)$ (rotational symmetry of three-dimensional Euclidean space). So this is a completely different concept that is only superficially similar and can't really be directly compared with orbital angular momentum. One way to see this is that spin can be a half-integer, but orbital angular momentum must always be an integer.
So to summarize:
Addition for the curious
As Eric has pointed out, there is more than just a superficial similarity between orbital angular momentum and spin. To illustrate the connection, it's useful to consider the question of how particle's properties transform under the change of coordinates (recall that conservation of total angular momentum arises because of the invariance to the change of coordinates that corresponds to rotation). Let us proceed in a little bit more generality and consider any transformation $\Lambda$ from the Lorentz group. Let us have a field $V^a(x^{\mu})$ that transforms in matrix representation ${S^a}_b (\Lambda)$ of the Lorentz group. Thanks to Wigner we know this corresponds to some particle; e.g. it could be scalar (like Higgs), bispinor (like electron) or vector (like Z boson). Its transformation properties under the element ${\Lambda^{\mu}}_{\nu}$ are then determined by (using Einstein summation convention)
$$ V'^a ({\Lambda^{\mu}}_{\nu} x^{\nu}) = {S^a}_b(\Lambda) V^b (x^{\mu}) $$
From this one can at least intuitively see the relation between the properties of the space-time ($\Lambda$) and the particle ($S$). To return to the original question: $\Lambda$ contains information about the orbital angular momentum and $S$ contains information about the spin. So the two are connected but not in a trivial way. In particular, I don't think it's very useful to imagine spin as the actual spinning of the particle (contrary to the terminology). But of course anyone is free to imagine whatever they feel helps them grasp the theory better.