Thanks so much for all your kind and also very interesting replies. Just now I was reading LONGMAN'S COMMON MISTAKES IN ENGLISH (2002) when I came across this phrase again (in Entry 399):
to make a bed (= to prepare the bed for sleeping on).
Then does 'make a/the/my bed' have another meaning (=turn down the bed)?
To "make a bed", or "make up a bed", is to put sheets, blankets, pillows and perhaps a top cover or eiderdown (duvet) on the bed so it is suitable for sleeping in. A properly made bed often has the top covers pulled all the way up over the pillows.
To "turn down a bed" is to partly turn down the cover/eiderdown to reveal the blankets underneath and then, as emsr2d2 says, usually turn down the corner of the blanket(s) and top sheet as well.
So, to make a bed and turn down a bed are not the same.
not a teacher