An object may be placed inside several collections simultaneously, when copying to/from collections Blender may pick the wrong one by default. Since these are not saved in any particular order the one picked may seem arbitrary to you. Cycling through the Index will allow you to specify which one to copy the name to. These are pre-sorted alphabetically for more predictable results.
Setting Collection Instance Offset from the menu is the main full featured operator.
From the properties window is a courtesy subset of functionality added for convenience. It only works for a single object at a time, and doesn't allow per-axis discrete control, or using the cursor simultaneously.
It can still save some steps of placing the cursor at an object before setting offset.
Since a collection is created for each selected object and centers are automatically set from coordinates of selected objects, it is a good practice to only select one object per desired instance collection this object should be the "main body" or "parent" representing the instance.
Having them all selected would lead to undesirable parasitic collections being created, and risk Instance Offset centers being misplaced.
But the whole point of instances is having multiple objects anyway, this defeats the hole purpose. How would I include several objects in said collection?
Ideally these secondary objects would be parented to this "main body" of the collection, and left unselected for the addon to deal with. You can also manually link them later with Ctrl+L > Make Links > Collections