This is not an answer, but it would probably be too long for a comment.
First some definitions and results I copied from Bartoszynski-Judah:
Definition 4.4.1: p-point = ultrafilter with pseudointersection property
Lemma 4.4.3: $\mathcal F$ is a p-point $\Leftrightarrow$ for every partition of $\omega$, $\{Y_n; n\in\omega\}$, either there exists $n\in\omega$ such that $Y_n\in\mathcal F$ or there exists $X\in\mathcal F$ such that $X\cap Y_n$ is finite for $n\in\omega$.
Definition 4.5.1: A filter $\mathcal F$ is called Ramsey if for every descending sequence $\{X_n; n\in\omega\}\subseteq\mathcal{F}$ of sets there exists a sequence $\{x_n; n\in\omega\}\in\mathcal F$ such that $x_n\in X_n$ for $n\in\omega$.
$\mathcal F$ is called a q-point if for every partition of $\omega$ into finite pieces $\{I_n: n\in\omega\}$ there exists $X \in \mathcal F$ such that $|X \cap I_n| \le 1$ for $n\in\omega$.
Theorem 4.5.2
Let $\mathcal F$ be an ultrafilter on $\omega$. The following conditions are equivalent:
$\mathcal F$ is Ramsey,
for every partition of $\omega$, $\{Y_n : n\in\omega\}$, either $Y_n \in \mathcal F$ for some $n\in\omega$ or there exists $X\in \mathcal F$ such that $|X_n\cap Y_n| \le 1$ for $n\in\omega$,
for every set $A\subseteq[\omega]^2$ there exists $X\in\mathcal F$ such that $[X]^2\subseteq A$ or $[X]^2\cap A=\emptyset$,
$\mathcal F$ is a p-point and a q-point.
When I tried to search in various literature (books, papers); I found quite frequently that Ramsey is equivalent to p-point and q-point. Most authors defined Ramsey ultrafilters using 3 (colorings). But I also found 2, sometimes under name selective ultrafilter.
I did not find the condition 1 in literature, the closest what I found was:
Lemma I.1.4. A nonprincipal ultrafilter $\mathcal U$ on $\omega$ is Ramsey iff for every sequence $\{M_i, i\in\omega\}\subseteq \mathcal U$ there exists $M \in \mathcal U$ such that $j \in M_i$ for all $i < j$ in $M$. (In Spiros A. Argyros, Stevo Todorcevic: Ramsey Methods in Analysis).
Several authors define selective ideals using diagonalization, which is very similar to condition from Lemma I.1.4.
Jech (Set Theory, Millenium Edition) in proof of Lemma 9.2 show as an auxiliary result that a decreasing system $X_n$ of sets from a Ramsey ultrafilter $D$ there exists $\{a_0 such that $a_0\in X_0$ and $a_{n+1}\in X_{a_n}$.
I believe that after replacing their definition of Ramsey ultrafilter by some of the above conditions, the proof from Bartoszynski-Judah might work; but I am still not convinced that their definition is incorrect.