$\text{MaxVote}$

$\text{MaxVote}(b, p, \mathcal{B})$

Give the previous constraints this function is defining the following, for a given Ballot id;
(Yes the b here is the Same as $B$bal, we're just getting started with his confusing notation)
a given priest/node and the set of ballots we will return vote with the largest ballot id of the set.

In words: this reads the vote from the set of votes in $\mathcal{B}$ cast by priest $p$ whose ballot number is less than $b$, or the null vote if none exist.

Conceptually, it returns a single vote (the one with the highest ballot number below $b$), not a set of votes or a ballot itself. This is a logical definition for the proof — it does not specify an implementation.

This also explains why Lamport switches notation from $B_{\mathrm{bal}}$ to $v_{\mathrm{bal}}$: he is referring to the ballot attached to the vote, not the ballot itself. The vote here is just a technical construct to access the associated ballot.

$\text{MaxVote}(b, Q, \mathcal{B})$

This function is an extension of the previous MaxVote function. Instead of taking a single priest/node it takes a set of them called a quorum. It then returns the maximum vote from the set of votes returned by calling the previous MaxVote function for each priest/node in the quorum.