In short. NO.
On the policy, item 1b reads "flights must be recorded by the Skywatch app". The term "recorded" used here might be confusing. The meaning of this section is that the insurance policy was properly filed (or put on record).
This simply means that you've successfully purchased a policy, got approval from the app and a valid copy of the policy to your email. After the policy was activated you may use any controller or app to fly and control your drone and record any footage you like, and still be 100% covered.