WASHINGTON—As the offshore oil industry’s federal regulator completed its overhaul of a major well-drilling safety rule in 2018, the agency’s director picked up the phone to a staff engineer to order up some changes.

Scott Angelle, director of the Bureau of Safety and Environmental Enforcement, told the engineer to delete language from memos showing that the changes would contradict guidance from the agency’s own engineers, according to emails and memos reviewed by The Wall Street Journal.

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