What is the correct time for fasting, what is the meaning of His saying, may He be exalted: {until the white thread of dawn becomes distinct to you from the black thread}, and what is the ruling on one who eats suhoor after the adhan, claiming that dawn has not yet become distinct?

The legal time for the commencement of fasting is the advent of true dawn, and the four madhhabs (schools of thought) have agreed upon this. One must abstain from things that invalidate the fast as soon as the dawn breaks, whether the muezzin has called the adhan or not. The criterion is the advent of dawn, not the adhan. If your relative had eaten suhoor after the true dawn appeared, then his fast is invalid and he must make it up. If it was before the advent of dawn, then his fast is valid.