TextEdit refuses to modify a file in /Users/Shared owned by another user despite mode 666
Notice that the permission string for /Users/Shared ends with a 't'. This indicates that the sticky bit is set for that directory. According to "man 8 sticky",


 A directory whose `sticky bit' is set becomes an append-only directory,
 or, more accurately, a directory in which the deletion of files is
 restricted.  A file in a sticky directory may only be removed or renamed
 by a user if the user has write permission for the directory and the user
 is the owner of the file, the owner of the directory, or the super-user.



I suspect that when TextEdit tries to save a file, it first attempts to rename or remove the old file. But in a directory with the sticky bit set, only the owner can do this. Hence, the permission-based failure.

You could test this by trying something like this as user Guest

echo " " &gt;&gt; /Users/Shared/Links.txt


If this is successful, it shows that Guest can write to the file and that TextEdit must be renaming or removing, not just rewriting the file.