How Can I Not Double Count Rows And Objects In My Query?

I am looking to count a number of instances for a group of people that had a request per month however I only want to count this once rather than however many times the person requested something. Here is an example of my data table so this might make more sense

Client    Customer ID   Request ID   RequestStartDate   RequestEndDate
 1         A1              9991            03/04/19         07/03/19
 1         A1              9992            07/05/19         08/05/19
 1         A1              9993            08/09/19         10/01/19

There are around 30k results but I wrote a query to grab the instances per month when a request was asked, this is what I came up with:

Select Distinct
CLient,
       (SELECT COUNT(*) from #Testdata21234 where '2019-07-01' BETWEEN dateadd(month, datediff(MONTH, 0, #testdata21234.RequestStartDate), 0) and dateadd(month, datediff(MONTH, 0, #testdata21234.BRequestEndDate), 0) AND CLient = g.Client) as 'July19',
    (SELECT COUNT(*) from #Testdata21234 where '2019-08-01' BETWEEN dateadd(month, datediff(MONTH, 0, #testdata21234.RequestStartDate), 0) and dateadd(month, datediff(MONTH, 0, #testdata21234.RequestEndDate), 0) AND CLient = g.Client) as 'August19',
    (SELECT COUNT(*) from #Testdata21234 where '2019-09-01' BETWEEN dateadd(month, datediff(MONTH, 0, #testdata21234.RequestStartDate), 0) and dateadd(month, datediff(MONTH, 0, #testdata21234.RequestEndDate), 0) AND CLient = g.Client) as 'September19'

FROM #testdata21234 g
group by client

My results end up like this if using the above example

July19    August19     September19
  2           2             1

I would like my output to get the following:

July19    August19     September19
  1           1             1

Essentially I want this to count once because although multiple requests, it is under the same Customer ID and I am not counting separate instances, just when the Customer actively pursued a request.

I hope this made sense and please, any help would be great.

Edit:

To further explain, lets try this example

Client   Customer ID   RequestID    RequestStartDate   RequestEndDate
1            A1          9991             03/03/19         07/03/19
1            A1          9992             07/05/19         08/05/19
1            A1          9993             08/09/19         10/01/19
1            A2          9994             07/02/19         07/10/19
1            A2          9995             07/15/19         08/06/19
1            A3          9996             02/01/19         07/01/19
1            A4          9997             08/05/19         08/16/19
1            A4          9998             08/30/19         09/01/19

For the above example, I need the outcome of the following counts by month

July19    August19    Septemeber19
4           3            2

I am counting based on clientID however need to filter based on a requestID. I hope this makes more sense, sorry for any confusion.

SELECT 
    FORMAT(g."RequestStartDate",'MMMM yy') month_year,
    COUNT(DISTINCT g."Customer ID") count_clients
FROM #testdata21234 g
GROUP BY 1

Select client,       
       count(distinct CASE WHEN RequestEndDate >= '2019-07-01'
                       AND RequestStartDate  < '2019-08-01'
                      then "Customer ID" end) AS July19, -- don't know if this matches your logic, once per customer?
       count(distinct CASE WHEN RequestEndDate >= '2019-08-01' -- overlap calculation
                       AND RequestStartDate  < '2019-09-01'
                      then "Customer ID" end) AS Aug19,
       count(distinct CASE WHEN RequestEndDate >= '2019-09-01'
                       AND RequestStartDate  < '2019-10-01'
                      then "Customer ID" end) AS Sep19
from #testdata21234
group by client;

Select client,       
       (select count(*)
        from #Testdata21234 g2
        where g2.RequestStartDate  < '2019-08-01' and
              g2.RequestEndDate >= '2019-07-01'
       ) July19,
       (select count(*)
        from #Testdata21234 g2
        where g2.RequestStartDate  < '2019-09-01' and
              g2.RequestEndDate >= '2019-08-01'
       ) Aug19,
       (select count(*)
        from #Testdata21234 g2
        where g2.RequestStartDate  < '2019-10-01' and
              g2.RequestEndDate >= '2019-09-01'
       ) Sep19
from (select distinct client #testdata21234) g;