$B$-meson naming convention

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 $ubar d$. Now the convention to distinguish between a meson and an anti-meson is that the negatively charged meson is the anti-particle, so the $D^+ = cbar d$ 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 $D^0=cbar u$ 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: $K^+=ubar s$. Then by extension you have $K^0=dbar s$, $bar K^0=bar d s$ 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.