$B$-meson naming convention An unbarred $B$-meson contains $\bar{b}$ (an anti-bottom quark), whereas a barred $\bar{B}$-meson contains $b$ (a bottom quark). What is the historical reason for this hellish naming convention?
 A: A meson is composed of a quark and an anti-quark.  For charged mesons in particular one quark will be an up-type quark ($u$, $c$, $t$) and the other one a down-type quark ($d$, $s$, $b$).  One of those will be a quark, and the other one an antiquark.  So the quark content of a $\pi^+$ is $u\bar d$.  Now the convention to distinguish between a meson and an anti-meson is that the negatively charged meson is the anti-particle, so the $D^+ = c\bar d$ is the particle, and the $D^- = \bar c d$ is the anti-particle.  Extend this to the neutrals (via isospin, if you will) and you find $D^0=c\bar u$ and $\bar D^0=\bar cu$, which is probably in line with what you expect.
Now let's look at the Kaons which contain the negatively charged $s$ quark: $K^+=u\bar s$.  Then by extension you have $K^0=d\bar s$, $\bar K^0=\bar d s$ which is contrary to your expectation -- but fairly immaterial as the $K^0$ and $\bar K^0$ mix to form the actually observed particles $K^０_L$, $K^0_S$.  Historically, strangeness was discovered before the quark model became established, so the strangeness $S$ for the $K^+$ meson was defined to be $S(K^+)=+1$ and thus, with our modern understanding $S = n(\bar s$ quarks$) - n(s$ quarks$)$.
I think you already understand what happens for the $B$ mesons: the $b$-quark is negatively charged so we follow the same scheme as for the Kaons.  Unlike the Kaons we can distinguish the $B^0$ and $\bar B^0$ experimentally, so we see the $b$ quark in $\bar B^0$  production and decays and $\bar b$ quarks in $B^0$ production and decays.  And thus we have to keep in mind that the $B$ particles contain $\bar b$ anti-particles and vice versa.
In short, remember this rule: the up-type quark content determines whether we consider the meson a particle or an antiparticle.  The historical reason is that strangeness predates the quark model.
