A representation of the Clifford algebra could be obtained by creation and annihilation fermionic modes $c_k$ and $c_k^{\dagger}$, by the following definition:
$$
\Gamma_{2k-1}=c_k+c_k^{\dagger}\,\,;
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\Gamma_{2k}=i(c_k-c_k^{\dagger})
$$
For $k$ being an integer between $0 <k< (d+1)/2$. For directions $\mu$ with negative signature you should put a imaginary unit $i$ up front. The Lorentz generator of
$$
\delta [...]^{\rho}=\frac{1}{2}\theta_{\mu\nu}(\eta^{\rho\nu}[...]^{\mu})
$$
where $\theta_{\mu\nu}=-\theta_{\nu\mu}$ is given by:
$$
\Sigma(\theta)=-\frac{i}{2}\theta_{\mu\nu}\Gamma^{\mu}\Gamma^{\nu}
$$
Note that for "rotations" in $(2k,2k-1)$ plane is given by
$$
\Sigma(\theta)=-\frac{i}{2}\theta_{2k,2k-1}(c_k+c_k^{\dagger})(i)(c_k-c_k^{\dagger})=\frac{1}{2}\theta_{2k,2k-1}(-c_kc_k^{\dagger}+c_k^{\dagger}c_k)=
$$
$$
=\theta_{2k,2k-1}\left(c_k^{\dagger}c_k-\frac{1}{2}\right)=\theta_{2k,2k-1}\left(n_k-\frac{1}{2}\right)
$$
This provides the interpretation of the modes. The modes are the eigenvalues the $\Sigma^{2k,2k-1}$ generator, being $+1/2$ for $n_k=1$ and $-1/2$ for $n_k=0$. The same as $J_3$ for the $d=3$ case.
For even dimensions, there is a chiral matrix commutes with the Lorentz generators (e.g. $\gamma_5$ for $d=4$) and can be written in terms of the modes as:
$$
\bar{\Gamma}=\prod_k(1-2n_k)
$$
This means that the eigenvalues of $\bar{\Gamma}$, the chirality, is $+1$ if there is a even number of occupied modes and is $-1$ for odd. Since this matrix commutes with Lorentz generator, for even dimensions we may split a Dirac spinor into two spinors, a chiral spinor ($\bar\Gamma =+1$) and anti-chiral spinor ($\bar\Gamma =-1$).
For odd dimensions one of this gamma matrices will be left over. For the last $k$-mode we take just the $\Gamma_{2k-1}$ and the $\Gamma_{2k}$ is unused. You may note that now there are two matrices that commutes with Lorentz generator, the $\Gamma_{2k}$ and the $\bar\Gamma$. It is important to note that they not commute with each other, so we can use just one of them to split the representation but not both. Whatever matrix we use we will end up with a reduction of the number of modes by one. If there is $n$ modes for $d=2n$, for $d=2n-1$ we have $n-1$ modes.
There is also other way to obtain the odd dimension gamma matrices for $d=2n-1$ by starting with $d=2(n-1)$ and using the $\bar\Gamma$ matrix as the $\Gamma^{d}$. This have $n-1$ modes from the start.
Now you can see that there is a distinction between odd and even dimensions. So, is always expected to have this jumps on the numbers of components when you reduce an even dimension $d$ to $d-1$, or increase an odd dimension $d$ to $d+1$, because of the increasing number of $c_k$ modes.
To answer the confusion between the three sources you list:
Using the signature convention of Figueroa O'Farrill, we have Majorana pinor representations for $p - q \pmod 8 = 0,6,7$ and Majorana spinor representations for $p - q \pmod 8 = 1$.
Pinor representations induce spinor representations (that will be reducible in even dimension) and so we get Majorana spinor representations for $p - q \pmod 8 = 0,1,6,7$.
Although $\mathcal{Cl}(p,q)$ is not isomorphic to $\mathcal{Cl}(q,p)$, their even subalgebras are isomorphic and so can be embedded in either signature. This means that Majorana pinor representations in $\mathcal{Cl}(q,p)$ also induce spinor representations in the even subalgebra of $\mathcal{Cl}(p,q)$ and so we also get an induced Majorana spinor representation for $p - q \pmod 8 = 2$
(from $q - p \pmod 8 = 6$; this is often called the pseudo-Majorana representation).
Fecko has his signature convention swapped compared to Figueroa O'Farrill, and so swapping back we see that his $0,2 \pmod 8$ gives us $0,6 \pmod 8$. One can also see from his table (22.1.8) that on the page you reference he was listing signatures with Clifford algebra isomorphisms to a single copy of the real matrix algebra, but his table also gives us $p - q \pmod 8 = 1$, converting signature convention to $p - q \pmod 8 = 7$ which is the isomorphism to two copies of the real matrix algebra and so also yields Majorana pinor representations. He doesn't talk about Majorana (or pseudo-Majorana) spinor representations here and so doesn't list $p - q \pmod 8 = 1,2$.
As for Polchinski, he includes pseudo-Majorana representations (or is signature convention agnostic) and so lists all of $p - q \pmod 8 = 0,1,2,6,7$.
To answer the question of in which dimensions Majorana spinors (including pseudo-Majorana) exist:
For a signature $(p,q)$ they exist whenever any of $\mathcal{Cl}(p,q)$, $\mathcal{Cl}(q,p)$ or the even subalgebra of $\mathcal{Cl}(p,q)$ are isomorphic to either one or a direct sum of two copies of the real matrix algebra. This means $p - q \pmod 8 = 0,1,2,6,7$.
If one discounts pseudo-Majorana spinors, then one removes $\mathcal{Cl}(q,p)$ from the previous statement and this means $p - q \pmod 8 = 0,1,6,7$.
Of course, this does not talk about the naturally quaternionic symplectic and pseudo-symplectic Majorana representations.
One can take the algebra isomorphisms of low-dimensional Clifford algebras ($\mathcal{Cl}(1,0) \cong \mathbb{C}$, $\mathcal{Cl}(0,1) \cong \mathbb{R} \oplus \mathbb{R}$ etc.) and use the isomorphisms between Clifford algebras of different signatures ($\mathcal{Cl}(p+1,q+1) \cong \mathcal{Cl}(p,q) \otimes \mathcal{Cl}(1,1)$ etc.) to bootstrap the equivalent matrix algebra isomorphisms of Clifford algebras (and similarly for their even subalgebras) of arbitrary signature and from there one can see when real forms exist.
Best Answer
In short: the spinning fields correspond to representations of the Lorentz group that we see arising for an isolated rotating body.
Internal angular momentum as Lorentz group generator
As a demonstration, consider a cloud of non-interacting particles in Minkowski space-time that you are viewing from afar. Let me use 3+1 splitted Minkowski coordinates so that I can define a Hamiltonian formalism and denote the coordinates as $t,x^i$ with $i,j,k=1,2,3$. Now each particle with mass $m_a$, $a = 1...N$ will have an on-shell 4-momentum $$P^\alpha = \left(\sqrt{m_a - \sum (P^i_a)^2}, P^1_a,P^2_a,P^3_a\right)$$ I also define the formal 4-position as $x^\alpha_a(t) = (t,x^i_a(t))$, where $t$ is simply the value of the time coordinate used to parametrize the motion of all the particles.
This Hamiltonian system naturally has the Poisson algebra $\{x^i_a,P_{ja}\} = \delta^i_j$ for every $a$ and otherwise all the variables commute. (Note that $P_i=P^i$ in Minkowski coords.) Now define an angular momentum tensor $$M^{\alpha \beta} = \sum_a (x^\alpha_a - x^\alpha_{\rm c.})P_a^\beta - (x^\beta_a - x^\beta_{\rm c.})P_a^\alpha $$ where all the functions on the right-hand side are given as functions of $t$, and $x^\alpha_{\rm c.} = (0,x^i_{\rm c.})$ is some referential centroid position (which is treated as Poisson-commuting with all the other variables here, for simplicity).
Finally, we define the total momentum $P^\alpha_{\rm tot.} = \sum_a P_a^\alpha$. Now we see that $P^\alpha, M^{\beta \gamma}$ fulfill the commutation relations of the generators of the Poincaré group (showing only non-zero brackets): $$ \{M^{\alpha \beta},M^{\gamma \delta}\} = \eta^{\alpha \gamma} M^{\beta \delta} + \eta^{\beta \delta} M^{\alpha \gamma} - \eta^{\beta \gamma} M^{\alpha \delta} - \eta^{\alpha \delta} M^{\beta \gamma} \\ \{M^{\alpha \beta},P^\gamma\} = \eta^{\alpha \gamma} P^\beta - \eta^{\beta \gamma} P^\alpha $$ where in this case you have to use the equations of motion to prove some of the identities. With a little bit of extra work you can see that these brackets will be the same in every 3+1 split so they can be considered as fully covariant.
Casimir elements and their relation to internal angular momentum
Now we see that there are a couple of Casimir invariants of the algebra such as $P_\alpha P^\alpha \equiv - m_{\rm tot}^2$ and $\mathcal{S} \equiv \sqrt{-M^{\alpha \beta}M_{\alpha \beta}/2}, \, \mathcal{S}^* = \sqrt{\epsilon_{\mu\nu\kappa\lambda}M^{\mu\nu}M^{\kappa\lambda}/2} $. The interpretation of $m_{\rm tot}$ as the total mass of the ensemble of particles (including their kinetic energy by $E=mc^2$!) is quite obvious. However, the interpretation of $\mathcal{S},\mathcal{S}^*$ is less obvious. We notice that in the 3+1 split we are using we can reparametrize the angular momentum tensor by the two 3-vectors $$ J^i \equiv \frac{1}{2} \epsilon_{ijk} M^{jk} \\ D^i \equiv M^{0i} $$ We naturally identify these as the mass dipole moment and angular momentum 3-vector with respect to the centroid $x^\alpha_{\rm c.}$. In terms of these 3-vectors we see that the Casimir invariants actually are $$ \mathcal{S} = \sqrt{J^2 - D^2} \\ \mathcal{S}^* = \sqrt{2\vec{J}\cdot \vec{D}} $$
Universality
Now, you have to trust me that such constructions are general enough so that this is the picture we get for any isolated classical system. Hence, general classical isolated bodies rotating around a center of mass will have non-zero $\mathcal{S}$ and possibly even $\mathcal{S}^*$.
Conversely, if $\mathcal{S}$ is real and positive for a massive isolated body, we see that the body will appear as having a non-zero angular momentum with respect to the centroid in every frame. Even more, since $M^{\alpha \beta}$ is the generator of Lorentz transformations and these Casimirs are all the Casimirs the algebra has, $m_{\rm tot}, \mathcal{S}, \mathcal{S}^*$ are the only properties of the ensemble that are invariant with respect to Lorentz transforms. When writing down coarse-grained (but still Lorentz-invariant) interactions of external fields with such rotating bodies, the numbers $m_{\rm tot}, \mathcal{S}, \mathcal{S}^*$ provide a universal set of parameters for the coupling.
Correspondence to irreducible representations
If you then switch to the quantum picture, you will see that the $(m,n)$ labels of the representations of the Lorentz group actually correspond roughly to the values of an alternative base of Casimir invariants $$ m,n \sim \sqrt{\frac{\mathcal{S}^2 \pm i (\mathcal{S}^*)^2}{2 \hbar^2}} $$ or $$\mathcal{S} \sim \hbar \sqrt{m^2 + n^2} $$ Now it is easy to see that nonzero $m,n$, such as $m=n=1/2$ for vector fields, must, inevitably correspond to a complex analytical continuation of the notion of an isolated rotating body with nonzero "internal" angular momentum.
(In fact, the highest weight of $SO(3)$ appearing within the $(m,n)$ representation would actually be $m+n$, so one can actually state that $J$ can be understood as reaching up to $\sim \hbar (m+n)$, depending on the choice of centroid. This justifies calling the $(m,n)$ fields as "spin $m+n$".)