Well, $B$, as defined in Besse on page 77, is not the Bochner curvature because it is not traceless when $m>1$ (cf. (2.64)). I believe that $B_0$ is some version of the Bochner curvature tensor.

Incidentally, the decomposition can be understood a little better by realizing that the space of curvature tensors of Kähler metrics at a point $p\in M$ is isomorphic to $P_{(2,2)}$ the space of real-valued polynomials on the complex vector space $T_pM$ that are homogeneous of bi-degree $(2,2)$. (Essentially, it's the degree $(2,2)$ part of the Taylor series of a Kähler potential function for the metric.) The metric itself at $p$ can be regarded as a positive definite real polynomial on $T_pM$ that is homogeneous of bi-degree $(1,1)$. Given $g_p\in {P^+}_{(1,1)}$, the space $P_{(2,2)}$ decomposes into subspaces as
$$
P_{(2,2)} = P^0_{(2,2)} \oplus \bigl(P^0_{(1,1)}\cdot g_p\bigr)
\oplus \bigl(\mathbb{R}\cdot (g_p)^2\bigr),
$$
where $P^0_{(k,k)}$ is the space of real-valued polynomials on $T_pM$ that are homogeneous of bi-degree $(k,k)$ and traceless with respect to $g_p$. This is the decomposition into the Bochner curvature, the traceless Ricci curvature, and the scalar curvature.

As for the normalizations in the formulas, you have to be careful with Besse because, in different parts of the book, different norms (all differing by constant scalar multiples, of course) are taken on the various tensor fields. For example, in some places, Besse takes the $2$-form $dx^i\wedge dx^j$ (on standard $n$-space) to have norm $1$, but, in other places, where it is regarded as $dx^i\otimes dx^j - dx^j\otimes dx^i$ embedded in the tensor product and thereby inheriting the 'natural' inner product, it is regarded as having norm $\sqrt2$. Make sure your normalization in your formulas agrees with Besse's when you are checking your constants.