The confusion here arises because we are not fully analogous to non-relativistic QM here.
Given a (quantum or classical) field $\phi$, we usually specify whether it is a "scalar", "spinor", "tensor", whatever field. This refers to a finite-dimensional representation $\rho_\text{fin}$ of the Lorentz group the field transforms in as an element:
$$ \phi \overset{\Lambda}{\mapsto} \rho_\text{fin}(\Lambda)\phi$$
But, simultaneously, the quantum field is an operator on the Hilbert space of the theory, and on the Hilbert space there must exist a unitary representation $U$. More precisely, every component $\phi^\mu$ of the quantum field is an operator, and hence transforms as operators do:
$$ \phi^\mu \overset{\Lambda}{\mapsto} U(\Lambda)\phi^\mu U(\Lambda)^\dagger$$
It is now one of the Wightman axioms that
$$ U(\Lambda)\phi U(\Lambda)^\dagger = \rho_\text{fin}(\Lambda)\phi$$
or, in components
$$ U(\Lambda)\phi^\mu U(\Lambda)^\dagger=\rho_\text{fin}(\Lambda)^\mu_\nu\phi^\nu$$
It is by this assumption that it suffices to give the finite-dimensional representation of the quantum field to also fix the accompanying unitary representation on the infinite-dimensional Hilbert space it is an operator on. The infinite-dimensional representations are characterized by Wigner's classification through their mass and spin/helicity. Since the finite-dimensional representations on the fields are also characterized by spins, the mass (from the kinetic term of the field) and the spin of the field (from its finite-dimensional representation) fix the unitary representation the particles it creates transform in.
All of this is often brushed under the rug because for the Lorentz invariant vacuum $\lvert\Omega\rangle$, we have
$$ \phi \lvert \Omega \rangle \overset{\Lambda}{\mapsto} \rho_\text{fin}(\Lambda) \phi \lvert\Omega\rangle$$
so knowing the finite-dimensional representation suffices to know how all states the field creates from the vacuum transform, and since the Fock spaces are entirely build out of such states, this is all the practical knowledge about the unitary representation that is usually needed.
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
The most famous theorem by Wigner states that, in a complex Hilbert space $H$, every bijective map sending rays into rays (a ray is a unit vector up to a phase) and preserving the transition probabilities is represented (up to a phase) by a unitary or antiunitary (depending on the initial map if $\dim H>1$) map in $H$.
Dealing with spinors $\Psi \in \mathbb C^4$, $H= \mathbb C^4$ and there is no Hilbert space product (positive sesquilinear form) such that the transition probabilities are preserved under the action of $S(\Lambda)$, so Wigner theorem does not enter the game.
Furthermore $S$ deals with a finite dimensional Hilbert space $\mathbb C^4$ and it is possible to prove that in finite-dimensional Hilbert spaces no non-trivial unitary representation exists for a non-compact connected semisimple Lie group that does not include proper non-trivial closed normal subgroups. The orthochronous proper Lorentz group has this property. An easy argument extends the negative result to its universal covering $SL(2, \mathbb C)$.
Non-trivial unitary representations of $SL(2,\mathbb C)$ are necessarily infinite dimensional. One of the most elementary case is described by the Hilbert space $L^2(\mathbb R^3, dk)\otimes \mathbb C^4$ where the infinite-dimensional factor $L^2(\mathbb R^3, dk)$ shows up.
This representation is the building block for constructing other representations and in particular the Fock space of Dirac quantum field.