When saving a filter from the Advanced search, it allows you to set the columns that the filter will have, and those columns can be seen by other users when selecting the "filter" section on the column selection.

When creating a search from the New issue navigator, you don't get that option, so there is no way to save the filter only with the columns you are interested in, instead, it saves the order of the columns you have added in the New issue navigator, but they will not be saved as the "filter" columns, they will just show all the fields (Default section).

If the search is saved as a filter, when we see the filter, we find that the fields selected in the New Issue Navigator are in the saved filter, and sorted in the same order:
But it also includes all the other fields selected in the "My default" section from the Advanced search, which is different than the fields shown in the My default section in the New Issue navigator:

It would also be good to have the columns saved in the same action that saves the filter, rather than changing the columns asynchronously like it is at the moment in the Advanced search (meaning that both the columns and the JQL filter would be saved through the same Save button).