"Early" and "late" nomenclature regarding O, B & K, M stars The Wikipedia article Stellar Classification has a subsection &quot;Early&quot; and &quot;late&quot; nomenclature.
It says:

Stars are often referred to as early or late types. &quot;Early&quot; is a
synonym for hotter, while &quot;late&quot; is a synonym for cooler.

My lecture notes say:

O and B stars are sometimes referred to as ‘early-type’, while K and M
are ‘late-type’.

If I look at a HR diagram, I see that main sequence O, B stars are hotter than main sequence K, M stars. So they are burning their fuel at a greater rate than K, M stars, so they run out of it early. So the stars which we are still able to observe now from the early universe are K, M stars, rather than O, B, which presumably formed rather *late**ly. Yet this is not consistent with the terminology above, so something is obviously wrong here.
What is wrong with my argument that the early and late words should be used in the other way round?
What's wrong with your argument is that you are using modern theories of stellar evolution. The terminology was introduced in the context of the theory that stars are powered by the Kelvin–Helmholtz mechanism (heat generated via contraction due to gravity). In this model, stars would start out hot and gradually cool down as they evolved, hence hot stars are called &quot;early-type&quot; and cool stars are called &quot;late-type&quot;.
This theory was shown to be inadequate because it can only power the Sun for tens of millions of years, while geological evidence indicated that the Earth had been around for far longer than this. The resolution was the discovery of nuclear fusion. However by this point the terminology of &quot;early-type&quot; and &quot;late-type&quot; stars had become established and it survives to this day.
(Incidentally the terms &quot;early-type&quot; and &quot;late-type&quot; are also used for galaxies, where once again they do not correspond to modern theories of how these objects actually evolve.)