"Early" and "late" nomenclature regarding O, B & K, M stars
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.)