Which airlines are the US flag carrier?

I think part of the confusion here arises from conflating "U.S. flag air carrier" with "U.S. flag carrier," which are two entirely different things.

What is a "Flag Carrier?"

A country's "flag carrier," is basically the either official or unofficial "national carrier" of a particular country. For example, Air France for France, KLM for The Netherlands, Lufthansa for Germany, etc. The U.S. does not have a flag carrier, though many decades ago (mostly during and before the 1960s) Pan Am was sort of an unofficial one. A country having a "flag carrier" is mostly a matter of national prestige, but can also be seen as a matter of national security (both military and economic) in the form of ensuring a country can maintain international air service under its own control.

For example, countries maintaining a "flag carrier" for national prestige is the main reason that there are still lots of different major international airlines in Europe (typically one per country,) even though nowadays almost all of them are owned by one of three companies in practice (IAG, AF/KLM, or Lufthansa Group.) The U.S. instead has 3 major international airlines (and a few other smaller ones,) none of which hold any special legal status compared to the others, so there is no "flag carrier" of the United States.

What is a "U.S. Flag Air Carrier?"

What they mean by "U.S. flag air carrier" for the purpose of the Fly America Act, though, is something completely different. Specifically, it means an air carrier that is "flagged" in the U.S. This means that the air carrier received its operating certificate from the U.S. FAA (which, in practice, generally means that the airline is based in the U.S. and typically also that their aircraft are registered in the U.S.) The U.S. Department of Transportation maintains a list of U.S. certificated air carriers on their website. Basically, though, the idea is that you must travel on an American airline unless that's not possible or your travel qualifies for one of the exceptions listed on the General Services Administration's website.

As far as the question about tickets operated by multiple carriers, GSA's website does address that part also:

Occasionally, two or more airlines will "codeshare" a flight by publishing and marketing the same flight under their own airline designators and flight numbers. You can purchase a seat on each airline’s designator and flight number, but the flight is only operated by one of the cooperating airlines. To comply with Fly America regulations, you must purchase the flight via the U.S. airline’s designator and flight number if the flight is shared between a U.S. and a foreign airline.

Note that there's a difference between merely purchasing a ticket sold by a U.S.-flagged airline and purchasing an actual codeshare. The important part here is the flight numbers that appear on the ticket. For example, if American Airlines sells a flight with an AA-coded flight number (e.g. AA6939,) that's ok even if the flight is actually operated by one of their partner airlines, such as British Airways. However, if the flight is sold to you by American Airlines, but with the flight numbers of one of their partner airlines (e.g. BA0178,) then that does not qualify, even though the flight is sold by American.

As mentioned in other answers, the source of US government grant funding may stipulate choice of a carrier according to the Fly America Act.

The Act specifies conditions where exceptions can be made - for example there is not a suitable US carrier, or it would take too long. But, as it says:

Note: Ticket cost and convenience are NOT exceptions to the Fly America Act.

However, the Act looks at the issuing carrier rather than the operating carrier. So it is fine to buy a ticket from American Airlines that codeshares onto a British Airways flight, but not the same flight bought from BA directly, nor for an AA flight ticketed by BA.

When searching for suitable flights you often have to be creative about codeshares or about stopovers. Using an advanced flight search (eg Google Flights or Matrix) can be helpful here.

Also things that aren't 'flights' aren't covered - such as trains and ships. Sometimes it can work out to fly into somewhere and then take a train for the onward journey.

A Flag-Carrier is a transport company that is registered as such in the given country (in this case the United States). They get preferential treatment by the government, and in many cases that includes things like being eligible for certain grants - especially when those grant monies come from the likes of the NSF and whatnot.

Your cited source notes that the Fly America Act restrictions are waived if you're flying to an EU member state, and are doing so aboard a flag-carrier for that state. So that's also an option, and that would meant that partnership routes like Iberian-American are valid... if you're going to Spain. Delta-Virgin would be legit to the U.K.

When in doubt check with your department's compliance folks before you book.