I agree you don't celebrate your 40th birthday at the beginning of your 40th year, though of course your 40th year commences on your 39th birthday and finishes on your 40th birthday.
So in terms of the millennium the 2000th year commenced at the beginning of 1999 and finished on the 31st December 1999 when the clocked clicked over to 2000.
Rather like filling up with fuel, the 20th litre commences once past 19.00 litres and finishes as the pump clicks from 19.99 litres to 20.00 litres.
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