So what is an American ship?
Simple answer: a ship flying the U.S. flag. But such a ship has to use an American crew, follow U.S. safety rules, and be built in a U.S. shipyard. That's very expensive.
So most ship lines go for "flags of convenience." These are the flags from countries that typically have no restrictions on what nationality your crew should be (or how much they should be paid) or where your ship should be built or repaired.
Here's the problem: the U.S. has a lot of government-related cargo moving around the world, like military supplies or disaster relief aid funded by U.S. tax dollars. Many folks, particularly members of Congress representing areas with shipyards and seafarers, believe those cargoes should move on U.S. ships. And so over the decades laws have been enacted to require just that.
But that has led to another problem: The U.S. does not have enough home-grown shipping capacity to handle all its cargo. Indeed, the handful of U.S. shipping companies in existence today only serve domestic routes (think Hawaii and Puerto Rico). There are no U.S. international shipping lines.
So how does the U.S. get its cargo around the world? Some foreign lines are allowed to fly the U.S. flags on portions of their fleet in exchange for a retainer and a guarantee of ship space when needed. And these carriers have to follow some of the U.S. rules, like crewing (think of the U.S. crew on the Danish line-operated ship Maersk Alabama in "Capt. Phillips").
In addition, there are some categories of cargo, typically aid or agricultural cargoes, where only a portion of the cargo has to move on U.S. flagged shipping. The rest can move on, well, ships with flags of convenience.
And here's an extra complication: Ships may be operated by one company, but chartered from another owner; much like the airline industry, where airlines lease out aircraft to meet seasonal traffic needs. Not to mention that many ship lines are publicly traded on overseas exchanges.
So a ship flying a flag of convenience (like the Marshall Islands) could have an operator that does a lot of business with the U.S. government (like Maersk), perhaps U.S. investors behind the scenes, and even U.S. controlled cargo on board…but still not be a U.S. ship.