Help on a proof about Ramsey ultrafilters

This isn't an answer, but it's too long for a comment.

Let's say that a non-principal ultrafilter $\mathcal{F}$ on $\omega$ is:

• Ramsey(BJ) iff for every decreasing sequence $X_n$ of sets in $\mathcal{F}$, there exists a set $\{ x_n | n \in \omega \} \in \mathcal{F}$ such that for each $n$, $x_n \in X_n$;
• Ramsey(BJ)${}^+$ iff for such a decreasing sequence $X_n$, the set of $x_n$ that we get satisfy the stronger condition that $x_0 \in X_0$ and $x_{n+1} \in X_{x_n}$;
• Ramsey(J) iff for every partition $\omega = \bigsqcup _n Z_n$ into "small" sets (i.e. each $Z_n \not\in \mathcal{F}$), there exists $X \in \mathcal{F}$ such that for each $n$, $|X \cap Z_n| \leq 1$;
• Ramsey(R) iff for every colouring $c : [\omega]^2 \to 2$ there exists $X \in \mathcal{F}$ such that $|c''[X]^2| = 1$; and
• Ramsey(R)${}^+$ iff for every colouring $c : [\omega]^n \to k$ (for any finite $n, k$) there exists $X \in \mathcal{F}$ such that $|c''[X]^n| = 1$.

The "BJ" is supposed to make you think of Bartoszynski and Judah, the "J" for Jech, and the "R" for Ramsey, since the Ramsey(R) definitions look most evidently like generalizations of Ramsey's theorem.

Now, Bartoszynski and Judah define a Ramsey ultrafilter to be a Ramsey(BJ) ultrafilter. And Theorem 4.5.2 in that text is essentially proving that Ramsey(BJ), Ramsey(J), and Ramsey(R) are equivalent (ignoring clause 4 of that theorem). The proof they give of "Ramsey(BJ) implies Ramsey(R)" seems to use Ramsey(BJ)${}^+$, however. So at first glance, there appears to be a problem.

On the other hand, Jech, as you might guess, defines a Ramsey ultrafilter to be a Ramsey(J) ultrafilter. Then in Lemma 9.2 he proves that an ultrafilter is Ramsey(J) iff it's Ramsey(R)${}^+$. He proves the forward (harder) direction in two steps, first he proves that Ramsey(J) implies Ramsey(BJ)${}^+$, and then that Ramsey(BJ)${}^+$ implies Ramsey(R)${}^+$.

So (assuming Jech's proof is correct, as is the rest of the proof in Bartoszynski and Judah), we know that the last four definitions above are all equivalent, and the first is weaker, but perhaps not strictly weaker. I doubt the definition given in Bartoszynski and Judah is a mistake (although I suppose it's possible), so how can we see that Ramsey(BJ) implies one of the other four characterizations?