Some of the War Galleys in the medieval era were really awesome, I think the king of that period was unquestionably the Dromon though. According to that article, the Dromon was used originally by the Byzantines only, and they developed variants of the design with more crew, like the Heavy Dromon you saw. But the design was eventually picked up by Arabs and other Mediterranean powers. Not sure if those designs had Greek Fire however. It was a closely guarded secret and there is a reason the fire siphon and greek fire dissapeared with the Ottomon conquest.
There was the War Cogs, I can't see a Quinquereme upgrading to that. I don't think they were even heavy ships, but they were more ocean-going than any Galley design. Can't say I'm a fan either. But the Medieval War Galley I like.
Renaissance was a pretty interesting period, start of age of sail and end of the long reign of the Galley. But the Galley didn't go overnight, you had Cannon armed galleys, Chebecks, which was half-way in design between a typical galley and later ships, more ocean worthy, and they evolved into a full sailing ship, Louis XIV had them in his fleet during the 100 years war. Then there was the heavier Galleas. Galleas and Cannon Galleys lasted up until sometime in the 1700s, during the hundred years war they were also used. The Renaissance was a cool period in maritime history I think.
Some later Dromon and Heavy Galley designs I found, a couple of these were images from models kept in a museum no less:
Looks like the hulls of these are transitioning away from the galley type hulls and more like a galleon, probably more ocean worthy? Probably got some tips from Venice or Genoan ship designs. Great example of some late Medieval warships though, no cogs here!
I think I see the flag of Venice on one of these.