At the risk of making things worse rather than better, let me point out that what appears to be one approach, say the Scott-Solovay version using complete Boolean algebras, is really a whole family of approaches, according to what universes you extend and how you extend them. Once you've defined a particular Boolean algebra $B$, you can do (at least) any of the following: (1) Build a cumulative hierarchy of $B$-valued sets, extending the usual cumulative hierarchy of ordinary (2-valued) sets. (2) Give a syntactic $B$-valued interpretation of ZFC in ZFC. [Intuitively, this is "the same" as (1), but technically it's a way of getting finitary proofs of relative consistency.] (3) Start with a countable transitive model $M$ of ZFC, interpret the definition of $B$ within $M$ to get an $M$-complete Boolean algebra $B^M\in M$, choose an $M$-generic ultrafilter $G$ in $B^M$, and form the countable transitive model $M[G]$, the smallest transitive model that includes $M$ and contains $G$. (4) Start with any model $M$ of ZFC (not necessarily well-founded), interpret the definition of $B$ in $M$ to get $B^M$, choose any ultrafilter $G$ in $B^M$ (not necessarily generic), and use it to form a 2-valued quotient $M^{(B^M)}/G$ of the Boolean-valued model $M^{(B^M)}$.
