Why in BCNS protocol both parties don't follow the exact same key extraction approach Why in BCNC key-exchange protocol (page 8), Bob instead of:

$K_B \leftarrow \lfloor \bar v \rceil_{2q,2} \in \{ 0, 1\}^n$

do this instead:

$K_B \leftarrow rec(2bs', c) \in \{0, 1\}^n$.

or they are the same thing ... am I missing something? I don't understand the purpose of modular rounding function as defined by Peikert in Lattice Cryptography for the Internet (page 9) and BCNS (page 6)

Because the idea is one party (i.e. Bob) sends a signal or reconciliation information (i.e. $c$) to other party (i.e. Alice) and then both should follow the exact same approach to get one bit from every coefficient.
Answer of your question is in page $7$.

Lemma2:

For odd $q$, let $v=w+e\in Z_q$ for $w,e\in Z_q$ such that $2e\pm 1\in E \pmod q$. Let $\bar v=dbl(v)$ then $rec(2w,\langle \bar v\rangle _{2q,2})=\lfloor \bar v \rceil _{2q,2}$.