since when setting up share permissions directly on my_folder I could see in Sharing tab the unc and description saying Shared.
I have no idea what you are doing. Are you setting up one share (shared_folder) or two (my_share) or three (my_folder)?
Now after modifying shared_folder share permission it works
This is what I tried to point out yesterday. You are describing things that you are doing, but not really providing any details. Forum users can only guess as to what permissions are set on any of these folders and any of the shares that you have.
In general share permissions act as a filter to the NTFS permissions, If a user has update access to D:\Data\Shared-folder but the share permissions to \servername\Shared-folder only have read access then the user will only have read access.
If you have D:\Data\Shared-folder\Sub-folder-1\Sub-sub-folder and wish to allow a user to access Sub-sub-folder through the share on shared-folder, then the user will need to at least have list access on Shared_folder and Sub-folder-1 in order to traverse the file system to get to Sub-sub-folder.