I did succeed in getting a stable header, i.e., one that is not
subject to unintentional, irreversible deletion.
Just wanted to report back and close this off.

The problem was that I wanted a different first -page header, and
2-column format. Apparently, when I set this up a couple years ago I
did it by putting a continuous section break at the beginning of an
empty document, then set the 2-column format to apply "from this point
forward" immediately after the section break. Most likely, I did it
that way because at that time the "header" was
not a proper Word header--- just a bunch of text boxes and a logo at
the top of the page, so i needed 1-column format there to make it
work. I later put the header material in a real Word Header, and
checked the box saying "different first page header,"
but did not remove the section break. The result was a two section
document with header-1 on the first page and header-2 on subsequent
pages. It worked, but when done that way the first page header is
rather volatile; if the section break is deleted suddenly there is
only one section so only one header is needed, and it is header 2 that
survives. Moreover, that action leaves the Undo on the Edit menu
grayed, so it cannot be undone. The header one material is lost
forever!

So, there you have it.

Thanks to all who responded. It was the interaction here that led me
to the solution.

Ed