The Constitution sets the minimum bar, but the legislature can grant citizenship beyond that.

I'm not sure whether it's due to a specific law, or courts applying 14A, but there's broad agreement that babies born in US airspace are, indeed, US citizens.

The jurisdiction language basically excludes parents with diplomatic immunity, under current rules, not other people flying above the US regardless of the aircraft's country of registration or any other status attributes of the parents.

In practice, what I said above is well-settled legal and administrative practice in the US, whether or not the constitutional words could have been interpreted or differently by Congress and/or the courts.

US authorities receiving proof of birth above the US and lacking reason to suspect parental diplomatic immunity certainly will and actually do grant proof of US citizenship, no further argument needed once the location of birth is documented.

ICAO disagrees, a child born on an aircraft is subject to the laws of the country the aircraft is registered in.

Although the registration country's laws do indeed generally apply on aircraft, that is in addition to what I said, not instead of it. Nationality laws can grant or withhold citizenship worldwide regardless of location and regardless of whichever other laws apply for most purposes.

For example, as an American who lived in the US for more than my first 30 years of life, any (biological) kid I have anywhere in the world will be a US citizen by birth, regardless of location, based solely on US law, even though outside the US I'm generally subject to the host country's laws and not to US laws. (Nationality laws are not the only US laws to have extraterritorial application, but the default rule is for US laws to apply domestically only.)

As an inverse example, the denial of US citizenship by birth to people born in the US to two diplomatically immune parents is solely due to US law, not due to any international agreement, even though the diplomatic parents are mostly immune to US law and subject to at least a large fraction of their home country's law.

Here's a general Wikipedia summary of how birth aboard aircraft and ships works for purposes of nationality, including specific comments on Canada and the US, as well as addressing the two international conventions that people often think are relevant to this question but aren't: https://en.wikipedia.org/wiki/Birth_aboard_aircraft_and_ship...

Very true, countries can deviate from this in the positive direction. So the US is free to grant citizenship to a lot more than these (e.g. anyone born with at least one US parent, that's also what my country does), but it's not required by ICAO, it's a decision the US can make unilaterally.

Right. But it's more than that: there's no default from which to deviate in any direction. Every country can freely choose who it considers as its citizens under all circumstances regardless of which law usually applies where, and I don't think any treaties cover that (with some narrow and not-worldwide exceptions around people who would otherwise be stateless) including the ICAO ones. ICAO's rule for the "Lex loci" of airplanes is irrelevant to the nationality question except when a nation's domestic law of nationality makes it relevant for granting (or withholding) that nation's nationality.

The US nationality rule for birth outside of the US is narrower than "anyone with one US parent", by the way. It wouldn't apply to cases like mine if my time spent living in the US had been below a certain threshold. By contrast, even children born in the US to those whose presence in the US is unlawful or otherwise irregular are citizens by birth.

Citizenship is defined by the country that grants it.