Where Groenendijk, Stokhof and Veltman (GS&V) say "peg", that translates in our terminology into a new "reference cell" or "location" in a store.

Where they represent pegs as natural numbers, that corresponds to our representing locations in a store by their indexes in the store.

Where they say "reference system," which they use the letter r for, that corresponds to what we've been calling "assignments", and have been using the letter g for.

Where they say r[x/n], that's our g{x:=n}, which we could represent in OCaml as fun var -> if var = 'x' then n else g var. (Earlier we represented assignments as lists of pairs, here we're representing them as functions. Either can work.)

Their function g, which assigns entities from the domain to pegs, corresponds to a store with several indexes. To avoid confusion, I'll use r for assignments, like they do, and avoid using g altogether. Instead I'll use h for stores. (We can't use s because GS&V use that for something else, which they call "information states.")

At several places they talk about some things being real extensions of other things. This confused me at first, because they don't ever define a notion of "real extension." (They do define what they mean by "extensions.") Eventually, it emerges that what they mean is what I'd call a proper extension: an extension which isn't identical to the original.

Is that enough? If not, here are some more hints. But try to get as far as you can on your own.