Get count of relationship rows between three tables that are not duplicates
I would need more information about your business restrictions to write an ideal query, but here's my advice given my interpretation of what you've said so far.

First thing you need to do is more clearly define what you are trying to achieve.


  So in cases where an article node is related to another node twice, both as a section and as a department, I would like to COUNT that only once.


Since your final result set will be grouped by departments and sections, "counting it only once" means you need to pick where they should be counted.  For example:

   content_field_department         content_field_section
+-----+----------------------+   +-----+-------------------+
| nid | field_department_nid |   | nid | field_section_nid |
+-----+----------------------+   +-----+-------------------+
|  1  |          2           |   |  1  |         3         |
+-----+----------------------+   +-----+-------------------+


should nid 1 be counted in the list for department nid 2 or counted for section nid 3?  Based on your question, you may want to only count it for departments.  In your current query drafts, you are trying to exclude the issue case from both counts, meaning you are effectively counting it 0 times instead of 1 (your goal) or 2 (what you are trying to avoid).

Here's a draft query as a starting point:

select
    DeptOrSect,
    nid,
    title,
    count(1)
from (
    select
        'DEPT' as DeptOrSect,
        dept.nid,
        dept.title,
        link.nid as article_nid
    from node dept
    join content_field_department link on link.field_department_nid = dept.nid
    union all
    select
        'SECT' as DeptOrSect,
        sect.nid,
        sect.title,
        link.nid as article_nid
    from node sect
    join content_field_section link on link.field_section_nid = sect.nid
    where not exists (
        select 1
        from content_field_department dept
        where dept.nid = link.nid
    )
) as x
group by DeptOrSect,
    nid,
    title;