Note: This is an edit of the previous question.

By $T_2$ conservatively extends $T_1$ if and only if:

There exists a function $F$ such that $T_2$ extends $T_1$ through $F$; and for every function $G$ such that $T_2$ extends $T_1$ through $G$, we have $T_2$ conservatively extends $T_1$ through $G$.

For definitions of the above terminology see "About conservative extensions of First Order theories?"

Now lets assume $T_1$ and $T_2$ to have the same language, like for example with the case of $ZFC$ and $Predicative \ MK$ set theories, where the later is a weakening of Morse-Kelley set theory by restricting class comprehension scheme to formulas in which all quantifiers are bounded in $V$. I think that $Predicative \ MK$ would conservatively extends $ZFC$ according to the above definition. The former theory is finitely axiomatizable.

Now my question is can we apply methods present in the answer to this question as to get also finitely axiomatizable conservative extensions [in the above sense] for all first order theories [meeting qualifications in that posting] even if they were written in the same language?

conservative extension, but afaithful interpretation. Any given theory has one, and only one, conservative extension in the same language, namely itself. You are confusing yourself by using sloppy terminology. If you choose to formulate ZF and this weakening of MK in the same language (as opposed to, say, formulating MK in a two-sorted language), then MK is not an extension of ZF at all, let alone a conservative extension. For example, ZF proves $\forall x\,\exists y\,x\in y$, while MK does not (it actually proves its negation). $\endgroup$ – Emil Jeřábek Mar 28 at 14:17