Detect answering machine and live human pickup is based on statistical data. No telephony system today is able to make it 100% accurate. This probably will never be possible. Even humans make mistakes in noisy environment like airport. But for most situations, the detection should work well.
Normally, the system will start the message after it hears the beep.
Your message might be played already but your voice mail is still waiting for the press of a key. If possible, you need to change your voice mail system prompt.