Speaking in depth for the first time about what he said was his role in the negotiations between the Russian president and Mr. Prigozhin, Mr. Lukashenko said that he had convinced Mr. Putin “not to do anything rash,” assuring him that “a bad peace is better than any war.” Mr. Lukashenko’s version of the conversation, reported by Belarusian state news media, could not be verified independently, and the Kremlin did not immediately comment on it.