Questions about semantics for First-Order Logic

The basic clause in the semantic definition of satisfaction for quantifiers in f-o logic can be stated in two alternative forms: (for simplicity I assume a formula A(x))

A) take an assignment function s that maps the set Var = { v1, v2, ... } of free variables into the domain D of the interpretation and consider the resulting truth-value of the sentence A(v1)[s]

B) take a name for each object c ∈ D, where D is the domain of the interpretation and consider the resulting truth-value of the sentence A(v1 / c̅).

Question 1) are there respectively : A) the objectual and B) the substitutional approach to quantification ?

Question 2) are the following the respective "correct" reading of them ?

for 1) Through s we assign a denotation (an object) to the term x (a variable) so that the formula becomes a sentence with a fixed meaning (i.e.it becomes meaningful);

for 2) I perform a substitution of a term (a linguistic entity) into the formula so that the formula becomes a sentence with a fixed meaning.

Question 3) The B) approach (substitutional) needs to be "corrected" (ref.BBJ, Computability & Logic (5th ed - 2007), pag.116), in order to take care of uncountable domains ?