Hi @Francesco Russo,
Yes. The description parameter and the Notes parameter are two different parameters.
I looked at the three admin centers as well as Outlook Client and PowerShell separately, and the same results as you. They cannot be synchronized and there is no way to modify notes parameters from the UI interface.
Therefore, I still recommend that you use set-group as a workaround to change the Notes parameters of a distribution group.
In addition, using set-group requires the user to have one of the following roles.
Here is a similar thread for your reference: How to edit a Distribution Group Note field in the GAL - Microsoft Community
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