I have a feeling the answer to this is either pretty obvious, or my assumption is really wrong, but I’ll take a shot at it anyway…
On a digital watch, which includes a stopwatch, you will have minutes, seconds, and two more digits. I think I have always heard those two digits referred to as “milliseconds”. I know that I have never heard the word “centisecond”. But if it is only two digits, shouldn’t it be something like centiseconds? Wouldn’t it have to be 3 digits to be a millisecond?
Can’t say as I’ve ever heard the term “millisecond” applied to the fractions of a second displayed on stopwatches. Race results and the like I’ve heard are usually given as “2 minutes, 3 and 12 hundredths (or sometimes 3.12) seconds.”
But I agree with your premise: those who use the term “millisecond” in that context are in error.
Concur
Milliseconds would also be pretty useless on a stopwatch that relies on a human pressing the button to start/stop it too, in fact, even the 100ths of a second are pretty much arbitrary because of human reaction times.