adding another count to this solution given

You say "if the month FALLS WITHIN the date range" [then show it].

A month is not a single day. It may "fall within" a range, but that would mean (to most English speakers) that the entire month falls within the date range. That's not the case for the examples you gave.

Rather, it seems that you want something else. If the month OVERLAPS WITH the date range, then show it. Is that correct?

If so - just to make sure we have a clear problem statement - if the range is 20 January 2022 to 14 February 2022, you want to show January 2022 (already by the old requirement, since the month-end falls within the date range), but ALSO show February 2022, since it overlaps with the date range?

Or is there a different, much simpler change that you are requesting now - if the date range doesn't include any month end (which, then, means that the date range is ENTIRELY within a single month), then show that month? Alas both examples in your post show a range that falls entirely in a single month.

To summarize, what is the desired result for the date range 10 January to 14 February? (Note that whatever your answer is, it SHOULD apply to the second row equally, assuming "null" is to be interpreted as "today".)

Here is how I understand the problem.

As before, I don't need to make any assumptions about time-of-day being midnight in all cases; that is just a special case. However, if in your problem it is important that times be treated as if they were all midnight, then you need some care with SYSDATE, which in almost all cases will have non-midnight time of day component.

If the time range ends, for example, on May 31 at midnight, that means at 00:00:00 on May 31. This range does NOT include the end of the month. The END_DT in this case is at the beginning of May 31; the entire day (except midnight at the beginning) is MISSING from the date range. So in this case, May overlaps with the date range, but the month end is NOT included. Note specifically that if in your test data (or even your real-life data) all dates are "pure dates", with time-of-day of midnight, then END_DT of May 31 means the month end is EXCLUDED. Is that what your end users require?

If the time range ends on June 1 at 00:00:00, then the end of May is included; and there is NO OVERLAP with the month of June, so it will NOT appear in the output.

If the time range ends on June 2, then the end of May is included, and the period overlaps with June but the end of June is not included.

Look at the three rows I added for testing, with ID = 5100, 5200 and 5300 respectively, and what I get in the output for those three rows. Is that what you need? If not, the query can be modified easily, but you must state very clearly what we should see for each input. And, clarify if your real-life data may have non-midnight time-of-day, how that should be handled, and how SYSDATE should be handled. (Do we need to use TRUNC(SYSDATE) everywhere?)

Anyway, based on my understanding of your problem, here is how I would change my earlier solution.

alter session set nls_date_format = 'Mon yyyy'; -- so I don't need to do it in the query

with
  sample_data (id, start_dt, end_dt) as (
    select 1001, date '2020-03-10', date '2020-12-31' from dual union all
    select 2005, date '2021-12-31', null              from dual union all
    select 3008, date '2021-04-10', date '2021-04-30' from dual union all
    select 4001, date '2021-05-12', date '2021-05-20' from dual union all
    select 5100, date '2021-05-03', date '2021-05-31' from dual union all
    select 5200, date '2021-05-03', date '2021-06-01' from dual union all
    select 5300, date '2021-05-03', date '2021-06-02' from dual
  )
select t.id,
       case when add_months(l.month_overlapped, 1) <= nvl(t.end_dt, sysdate)
            then l.month_overlapped end as monthend_included,
       l.month_overlapped     
from   sample_data t
       cross apply
       (
         select     add_months(trunc(start_dt, 'mm'), level - 1) as month_overlapped
         from       dual
         connect by add_months(trunc(start_dt, 'mm'), level - 1) < nvl(end_dt, sysdate)
       ) l
;

    ID  MONTHEND_INCLUDED  MONTH_OVERLAPPED  
------  -----------------  ------------------
  1001  Mar 2020           Mar 2020          
  1001  Apr 2020           Apr 2020          
  1001  May 2020           May 2020          
  1001  Jun 2020           Jun 2020          
  1001  Jul 2020           Jul 2020          
  1001  Aug 2020           Aug 2020          
  1001  Sep 2020           Sep 2020          
  1001  Oct 2020           Oct 2020          
  1001  Nov 2020           Nov 2020          
  1001                     Dec 2020          
  2005  Dec 2021           Dec 2021          
  2005  Jan 2022           Jan 2022          
  2005  Feb 2022           Feb 2022          
  2005  Mar 2022           Mar 2022          
  2005  Apr 2022           Apr 2022          
  2005                     May 2022          
  3008                     Apr 2021          
  4001                     May 2021          
  5100                     May 2021          
  5200  May 2021           May 2021          
  5300  May 2021           May 2021          
  5300                     Jun 2021         

Sorry, but between this quote and the other thing you mentioned (that the dates may have non-zero time-of-day and that the time of day must be considered), this makes absolutely no sense to me.

A period from May 3, 8:30 AM until May 31, 4:45 PM does not overlap the end of the month. The end of the month is at midnight at the END of May 31, and your period ends strictly before that (more than 7 hours before it).

Perhaps in real life, your users told you that the period does overlap the end of the month if it overlaps 00:00:00 (12 AM) of the month-end date, but that doesn't mean it makes sense. I give up. The query can be modified very easily to implement all your clarified requirements (including this one), but I choose not to work on problems that don't make sense to me.

Good luck with your project!