Exogenous Money Supply In the first 10 seconds of the following video: https://youtu.be/anZ58gZcxqk,
A claim is made that in the Exogenous Money Supply Model, Money Supply is not determined by interest rates. The person further adds that the money supply is independently determined by the central bank.
My question is that isn't the central bank only responsible for determining the interest rates? I know there are other tools such as open market operations that can be used to control the money supply in an Economy. But, If the central bank is determining the money supply, shouldn't it then be a function of interest rates.
My question is that isn't the central bank only responsible for determining the interest rates?

No, central banks do not even have directly responsibility to adjust interest rate. Depending on which country we talk about they will have responsibility to keep prices stable, high employment and bank regulation and often combination of the above. In order to follow their mandates they will often fiddle with interest rates but typical central bank has many instruments at its disposal. For example, they can also increase money supply or even adjust reserve requirements that also affect money supply (Fed recently set them to zero in their effort to increase inflation).
As mentioned above  most of the time contemporary central banks adjust interest rates on their reserves that influences interest rate in the economy and let the new money supply be determined by what the interest rate is but I don't know of any central bank that would not be allowed to directly increase $M$ if they would want to in some way. Also, note even if central bank sets itself some goal achieving some interest rate without stating any goals in terms of money supply, since interest rates depend on what $M$ is, they in principle could fully control interest rate just by varying $M$. They might want to use more tools because relying only on $M$ would be more difficult but it is possible.
In fact the open market operations that you mentioned are one way how central bank can directly increase money supply in the economy and the open market operations are not that scarce. These are becoming increasingly common, for example you can see from the graph below what an effect QE (which is not fundamentally much different from OMO expect in its scale and that it is part of holistic strategy that includes other tools) had on increase in monetary base in the US (first round of QE started in 2009 you can easily spot it by the shar increase in monetary base), so that is non-trivial  change in $M$ (technically $M0$ but $M$ consist of base money as well) that wasn't created with interest rates.