Almost-P and related definitions
The question seems to be predicated on a misunderstanding: the statements “relative to a random oracle $A$, $\mathrm{P}^A=\mathrm{BPP}^A$” and “$\mathrm{Almost\text-P}=\mathrm{BPP}$” are not meant to be rephrasings of each other. The complexity zoo refers to a paper of Bennett and Gill, which proves the former statement (and many other things) in detail, but it also separately claims the second statement (though they do not use the $\mathrm{Almost\text-P}$ notation, and do not really give a proof).

The definition of $\mathrm{Almost\text-P}$ in the zoo is correct.

The proof of $\mathrm{Almost\text-P\subseteq BPP}$ goes as follows. Let $L\in\mathrm{Almost\text-P}$. Since there are only countably many Turing machines, there exists an oracle polynomial-time Turing machine $M$ such that $M^A$ computes $L$ with positive probability $\epsilon&gt;0$. Using the Lebesgue density theorem, there exists a finite oracle prefix $A_0$ such that relative to random oracles $A$ that extend $A_0$, $M^A$ computes $L$ with probability $\ge3/4$. We can hardwire $A_0$ into the Turing machine, and simulate access to the rest of the oracle by random coin flips to obtain a randomized poly-time machine for $L$ with probability of success $\ge3/4$. Thus, $L\in\mathrm{BPP}$.