Well... the genius who was working at the counter in the (very famous) store where I bought my new Yamaha 211 silver plated flute, insisted to show me how to move the cork inside the headjoint in order to get the perfect intonation. He took the plastic cleaning rod and forced it inside the headjoint
in order to move the cork. As a result, he scratched visibly the silver plating on the round plate that covers the end of the cork inside the head joint. Later, while reading the Yamaha care manual, they suggest to put a cotton fabric around the cleaning rod when using it to move the cork, precisely in order not to damage the silver plate. My question is: this is an area that gets a lot of wet... will the plate that covers the cork become rusty in the future ?
Should I insist that it is replaced ? Or is it just a few scratches with no problem ? Thank you for your opinion.

No doubt it has to be replaced.
But in fact, you should never use the plastic or wood stick to move the cork, because you use a small portion of the total area of the metal, and although this is a pretty hard metal, it could deform.
Get a wood round stick, aproximately the diameter of the smaller end of the headjoint, with one of its end perfectly flat, so you make full contact to the metal part of the stopper when you need to move it.
Another aspect of the cork is that you generally move it once to put it in the right place, and you do not need to change any more, unless you replace it.
The key here is that the cork must be sufficiently "squeezed" between the two metal plaques so it doesn't move unless you want it to.
Remember to spread a small amount of cork grease in the cork surface so it avoids air to scape.

'will the plate that covers the cork become rusty in the future?' No because they are made of nickel silver or brass not steel.

The most common cause of corks getting moved is that the crown (The silver 'stopper' at the end of the flute) comes unscrewed & players screw it down as hard as they can which moves the cork further down the head. This causes the flute to play flat, especially the higher registers. Also it compresses the cork so that it might be too loose when put back into position & won't stay in position. Also there is a danger that air leaks will occur. The other major cause of loose corks is age - they shrink with the same problems as before.

I looked at the plate with a small torch and it does not seem deformed to me, just scratched.
Should I ask the store to replace it any way ? Or should I give up, knowing that it will not rust ?

A few scratches on it are not going to hurt anything. I see student flutes all the time that have scratches there because of metal cleaning rods that have hit it when cleaning it by kids. It is no biggy. The person at the store most likely was trying to explain (but not correctly) that there is a line on the end of the cleaning rod that when inserted in the headjoint, it should be seen in the very middle of the embouchure hole. You do not move the cork around for tuning a flute. It should stay in that one position. If it moves, then the cork needs to be replaced as it is then too loose.

Thank you for your answers. Yes the counter guy was silly because he really wanted to move the cork (!) to show me how to correct the intonation of a flute...
and he pushed hard because the cork was not moving... so he made the scratches on the surface of the silver plate that covers the cork internally. You know, I am a beginner and this was my first flute, just bought brand new... to see that guy doing that was painful ! But I am glad to be told that it's not a big deal. Now I should focus less on those scratches and more on learning to play !
Thanks again.