The $\Delta$-system Lemma states that given an uncountable family $\mathcal C$ of sets with finite intersection there exist an uncountable subfamily $\mathcal D$ of $\mathcal C$ and a finite set $\Delta$ such that $d_1 \cap d_2=\Delta$, for all $d_1$ e $d_2$ distinct elements of $\mathcal D$. Does MA($\kappa$) imply that given a family $\mathcal C$ of $\kappa$ sets with finite intersection, there exist a subfamily $\mathcal D$ of $\mathcal C$ with $|\mathcal D|=\kappa$ and a finite set $\Delta$ such that $d_1 \cap d_2=\Delta$, for all $d_1$ e $d_2$ distinct elements of $\mathcal D$?
Martin's Axiom plays no role here.
This generalization of the Delta System Lemma is true (in ZFC) if and only if $\kappa$ is a regular uncountable cardinal. This is lemma III.2.6 and exercise III.2.7 in Kunen's book Set Theory (the last edition).
Edit: As Mike pointed out in his answer, there is a missing assumption: one must assume that the family consists of finite sets (both for the usual lemma and its generalization).
There are also generalizations for family of infinite sets. See for instance the answer from Joel David Hamkins to this question.
Your formulation of delta-system lemma and attempted generalization (under MA) fail because there is an uncountable almost disjoint family of infinite subsets of $\omega$. You must assume that your family consists of finite sets and not just that their pairwise intersections are finite.
I am not sure what Charles' answer was intended to address.