Robert S. Mueller III, the special counsel investigating Russia’s interference in the 2016 election, transformed this week what has appeared to be a weakness — his relative lack of independence — into a strength.

Since he was appointed in May by Rod J. Rosenstein, the deputy attorney general, the scope of Mr. Mueller’s power has been intensely debated. President Trump’s allies have portrayed the special counsel as running amok as his focus expanded to the business dealings of Mr. Trump’s associates. The president’s critics have worried that Mr. Mueller is too vulnerable to potential Trump administration interference.

The tension is familiar when the president comes under scrutiny by law enforcement, as the vast powers of the executive branch are at stake. Similar tensions cropped up during the Watergate, Iran-contra, Whitewater and Valerie Plame leak investigations. Now they have spurred the first significant legal fight arising from Mr. Mueller’s work, as Paul Manafort, Mr. Trump’s onetime campaign chairman, battles charges related to payments he received from a pro-Russia government in Ukraine.