I think this is a good analogy. The UK, as it is set up now, has been called a regime of asymmetric federalism: It is split into four countries (England, Scotland, Wales, and Northern Ireland), and three of these four countries have their own legislatures. It’s not exactly true federalism as in the US, because in true federalism, the component states possess their powers inherently by virtue of being states, whereas in the UK, these legislative powers have been delegated (devolved) to the component countries by the central Parliament; but it’s close enough, in the sense that Scotland, Wales, and Northern Ireland have their own parliaments with the power to legislate autonomously within defined areas of competence.
England does not have such a legislature; those matters which are, in Scotland, Wales, and Northern Ireland, under the control of the respective legislatures, are, for England, governed by the UK Parliament - including the MPs from Scotland, Wales, and Northern Ireland. To put it bluntly: Scottish voters have a say on internal matters of England, but English voters do not have a say on the same internal matters for Scotland.
This has parallels to the District of Columbia. It is subject to the control of the national parliament (Congress), including those members of Congress elected in, say, Texas, but the DC voters do not have a corresponding say in the same matters for Texas.
So would it be correct that the US President is the “President of the District of Columbia”? It is, in the same sense it is correct to say that Elizabeth II is the “Queen of England”, and by the same arguments as those we’ve read in this thread. The US President resides in DC, holds powers in DC, is the head of state and government in the country of which DC is a part, and there is nobody in DC who could otherwise be referred to as its President. But do we call him the President of DC? Hardly.