I suspect that the Cuenca call is based on looking at the pictures of the hat; it appears to be the brisa weave, which is different from the llano weave of the hats more often called Montecristi hats (although there are probably some brisa weave hats made in the Montecristi area). Do a search for brisa llano, there have been lengthy discussions about this.
Both are panamas, and there are discussions about this also, that just about all straw hats of any weave made in South America were being shipped out of the Panama region during construction of the Panama Canal, and newsreel/newspaper coverage popularized the 'panama' hat. The brisa is an easier weave, uses wider straw strands, lower weave count. The llano can have a lower weave count, but I think that the prized higher weave counts are only the llano weave Montecristi hats, typically woven by a relatively few weavers near the village of Pile.
I am remembering from previous threads here, but there are several regulars here that have more direct expertise, hopefully they will clarify any errors I made above.