  We characterize the big Ramsey degrees of free amalgamation classes in finite
binary languages defined by finitely many forbidden irreducible substructures,
thus refining the recent upper bounds given by Zucker. Using this
characterization, we show that the Fra\"iss\'e limit of each such class admits
a strong big Ramsey structure, implying that the automorphism group of the
Fra\"iss\'e limit has a metrizable universal completion flow.
