(1) $T = \{(\forall x) (x=c_1 \lor x=c_2 ) \}$
(2) $T =\{c_1\neq c_2 \}$,
$T' = \{(\exists x)(\exists y)((x \neq y) \land (\forall z)(z=y \lor z=x))\}$
I am not sure, how to determine that T' is or isn't an extension/conservative extension of T.
In first theory, there is an axiom that is valid in structure, which has at most 2 elements. T' is valid for structures, which universe has just two elements. So I think the T' cannot be extension of first theory, because T' axiom cannot be valid in structures, which has one-element universe.
In the second example there is a theory, which axiom is valid in only two-element structure. Both theories are valid in same models, so T' is extension of theory T.
Extension is conservative, if we can prove any formula in language of theory T from axioms of theory T'. I would say, it si conservative extension, because we can prove any formula in language of theory T, which has two constants, in theory T'.