You are confused about the way how the dimensions of representations are counted. In general, they are complex dimensions, not real ones.
More precisely, representations of Lie groups come in three types: the complex ones, the real ones, and the pseudoreal ones. The complex ones are inequivalent to their complex conjugates. The real and pseudoreal ones are equivalent to their complex conjugates and the real ones are such that the equivalence may be used to demand that the coordinates of the representations are real. The pseudoreal or "quaternionic" representations can't be reduced in this way but they're still equivalent to their complex conjugates essentially because $i$ and $-i$ may be continuously connected with one another through the sphere $S^2$ of the unit, pure imaginary quaternions.
$SO(4)$ is locally isomorphic to $SU(2)\times SU(2)$ so it has two inequivalent 2-dimensional complex (pseudoreal) representations. One of them is doublet under the first $SU(2)$ and invariant under the second $SU(2)$ transformations, the other is a doublet under the other $SU(2)$.
Similarly, $SO(5,1)$ is a sort of $SL(2,H)$ group of $2\times 2$ matrices with quaternionic entries whose "real part of the determinant" equals one. This group may be written in terms of $4\times 4$ complex matrices, so it is a subgroup of $GL(4,C)$. It follows that $SO(5,1)$ has 4-dimensional (complex, well, pseudoreal) fundamental representations. Well, it has two inequivalent pseudoreal fundamental representations and they're not complex conjugates to each other. Instead, each of them is equivalent to the complex conjugate of itself.
So only for real representations, the counting of the dimensions follows what you believe. Complex representations that don't admit any natural "restriction making the coordinates real" (without doubling) are complex and by the dimension, we mean the number of complex coordinates (without any multiplication by two). Representations with $K$ quaternionic coordinates are counted as $2k$-dimensional complex representations. This is the unified way to treat representation that leads to a more uniform and regular set of rules to determine how things behave. This is the most natural way because it's based on complex numbers and complex numbers are more fundamental than the real numbers or quaternions. (Fundamental theorem of algebra and other reasons.) Real and quaternionic representations are classified as complex representations with the special freedom to "conjugate" coordinates, i.e. with some special "antilinear structure map $j$" that commutes with the action of the group $g(v)$. For real reps, $j^2=+1$, for pseudoreal ones, $j^2=-1$ and $j$ may be literally interpreted as the multiplication by the $j$ quaternion from the proper side.
Around B.1.43 and B.1.44, Joe simply tells you to diagonalize the representations on both sides and list possible eigenvalues of all the operators $S_a$ - look at the basis of the representation containing all the shared eigenstates of all the $S_a$ operators. All these eigenvalues of $S_a$ are $\pm 1/2$ - the collection is known as the weight. Whether the number of the negative $-1/2$ eigenvalues is even or odd decides about the chirality of the spinor.
So the left hand side of B.1.44 are the collections of weights (eigenvalues under $S_a$ operators) that are $\pm 1/2$ each and the number of the negative eigenvalues is even (a) or odd (b). They may be obtained as tensor products of collections of smaller sets for which the chiralities are even for the left group and even for the right group or odd for the left group and odd for the right group (a), or even-odd or odd-even (b). This is why the irreducible rep of the larger group decomposes into the direct sum of two irreducible reps of the factor groups and each of the two terms in the direct sum is a tensor product of two Weyl spinors.