WASHINGTON—The White House moved to clamp down on access to records of presidential phone calls after problematic leaks of sensitive discussions early in the Trump administration, according to people with knowledge of the actions, culminating in the use of a secret national security server which is now central to the impeachment probe.

Back-to-back leaks of controversial remarks by President Trump during calls with leaders of Mexico and Australia resulted from an unusually loose record-distribution policy in the first days...