Hello,
I think I got it.
It had to do with security, but not at all the way I thought of it.
The project had been created in an administrator session.
Later, to make modifications, I accessed it from a limited user session,
with identity delegation from the administrator, thanks to either the
runas command, or a shortcut with appropriate properties.
It appears HTML Help Workshop does not support Identity Delegation that way.
So, let us describe what happened.
Just after having modified the page and compiled the help file, I opened
it, saw the modifications were done, closed it and went to something else.
I had modified the text, I saw no reason to test all the buttons in
detail again.
This was an error. Most of the buttons on the modified pages did not
work any more, and the pages that were supposed to be opened in pop-ups
were opened in the same window as the link.
So, today I closed the limited user session, went to the admin session,
where I verified my modifications and compiled the help file, and
everything was running fine again.
In fact, I first created a copy of the source thanks to the
decompilation of a copy of the help file that worked, but eventually
this was just useful to help me understand what happened, and that a new
compilation in the good session (and of course a deployment afterwards)
was the solution.