Hello, I realize this is an older thread however in my search to accomplish the same thing I found a way that worked for me at least. On my public mapshare page I was not able to delete tracks I had recorded with my new inreach mini that I was testing to get to know the unit.
I had to go to garmin.explore.com and login. When I first set up the inreach mini it automatically created a collection "sent to mapshare". I found the track in there and selected it. A bin symbol to delete shows up there in the upper right had corner when you expand the track. I pressed this and the track entry in the column disappeared but the track stayed on the map and was selectable. From there I found the track again in my "unorganized" folder in the library and was able to select and delete that. The track then disappeared off the map and is now gone completely on my mapshare page.
When this thread started, there were no collections.
To be clear, no data actually exists IN a collection. All the real data exists in the library. Collections are just "pointers" into the library data. Collections are mostly used to control what is sync-d between the device and the Explore web site (oversimplification there).
MapShare can tell the difference between a random visitor and the MapShare page owner - as long as the owner is logged in when visiting MapShare. That said, I am not sure that it would allow you to delete anything from there. The safest thing to do (in so far as anything is safe) is to manipulate ALL your data from explore.garmin.com - not from MapShare.
Personally, I handle hiding things on MapShare by setting the "earliest date shown". It's easy and (the last time I looked) reliable for hiding things without destroying data.
Deleting the track from library is equivalent to deleting it from the explore web site in the old days. When you delete it from the library, it is gone. References to it will be automatically removed from any collections as well.
This still leaves the data on the device itself. I suspect that it will reappear on the web site the next time you sync. It IS a bi-directional sync. But the originator of the data (in this case, the Mini that recorded the track) takes precedence.
Bottom line: If it comes back after the sync, delete it from the device AND from the library before you sync again.