The statement on the learncpp.com website is correct in principle, and your experimental results, while seemingly contradictory, can be explained by a technical detail of C++ streams and how they are typically configured.
The primary reason you observed std::cerr to be faster than std::cout is most likely due to a feature called synchronization with C standard I/O. By default, C++ streams (std::cout, std::cin, etc.) are synchronized with their C counterparts (stdout, stdin, etc.). This synchronization ensures that if you mix C and C++ I/O functions (e.g., printf and std::cout) in the same program, the output will appear in the correct order.
This synchronization, however, comes with a significant performance penalty. It effectively forces a flush of the std::cout buffer after every operation to ensure all data is immediately written out. In this default configuration, std::cout behaves more like std::cerr in that it does not fully utilize its buffer for performance.
Since std::cerr is an unbuffered stream by default and is not tied to stdout in the same way, it bypasses this synchronization overhead, which is why it appeared faster in your tests.
The statement from learncpp.com is based on the general principle of I/O buffering. When you disable the synchronization, std::cout can buffer its output, allowing the operating system to write a large chunk of data to the console in a single, efficient operation, which is much faster than the numerous, individual write calls required by an unbuffered stream like std::cerr.
For a program that performs a very large number of I/O operations, such as a competitive programming solution or a data processing script, you can dramatically increase the performance of std::cout by adding the following line at the beginning of your main function:
std::ios_base::sync_with_stdio(false);
This line disables the synchronization, allowing std::cout to fully utilize its buffer and perform significantly faster than std::cerr for bulk output.
Your program was not complex enough to demonstrate the full performance potential of std::cout because the default synchronization setting was hindering its performance. The experiment you conducted highlights a common performance trap for C++ programmers who are new to the language.
C++ Weekly: Use cout, cerr, and clog Correctly
This video explains the correct use of std::cout, std::cerr, and std::clog.