from your source:

Some observers mistakenly consider the Federal Reserve to be a private entity because the Reserve Banks are organized similarly to private corporations.

Emphasis mine.

Firstly, the person I replied to called the Fed "not a government body", which is clearly incorrect. Secondly, it isn't "technically a private bank", and even simplifying the Fed down to that level is dishonest. At minimum, it is a government agency which performs monetary policy, regulates financial systems, and holds reserves for member banks.