Is the "anonymous proxy server" I came across on the Internet really an anonymous one?

Most probably, No. We have studied more than 8,000 public proxy servers and found that only a few hundred of them are true anonymous proxy servers. Although almost half of all servers call themselves "anonymous" because they do not transfer the client's IP address in the standard way, nevertheless the address is present in their requests in other different forms and therefore they do not provide any anonymity. Besides, we found that some proxies transfer the IP address of the client occasionally, from time to time.