If you drag a link or favicon on the left end of the location bar onto the desktop to create an internet shortcut then that shortcut gets the icon of the default browser.
If you want a different icon (favicon) then you have to assign that icon yourself to the desktop shortcut (right-click: Properties).
You can usually get the favicon if you append "favicon.ico" to the main domain of a website (e.g. http://www.mozilla.com/favicon.ico ) to display the favicon in a tab and save that image to a folder or see if you can find it in "Tools > Page Info > Media".

why do think the answer resolves the problem. did u understand the question. you have answered the wrong question.

when i create a shortcut i want to have the web site's icon not the one of the default browser ... why do we have different icons ? to distinguish ... so we do NOT expect the default browser to destroy it.

where this amazing smartness comes from ? why cause damages .. easier than thinking correctly ?

That is the way Windows works with addon web browsers. cor-el provided instructions for overcoming that shortcoming.