Do You Stop Eating at Fajr or Sunrise?
Question

What is the ruling on eating or drinking during the adhan for Fajr? The Prophet (peace and blessings of Allah be upon him) said: “If the iqama for prayer is given and the vessel is in the hand of any one of you, let him not put it down until he has finished with it.”
Praise be to Allah.Hadith about eating during adhan in Ramadan
The hadith mentioned in the question was not narrated in that wording. The wording of the hadith is: 
Abu Hurayrah (may Allah be pleased with him) said: The Messenger of Allah (peace and blessings of Allah be upon him) said: “If one of you hears the call (to prayer) and the vessel is still in his hand, let him not put it down until he finished with it.” (Narrated by Ahmad, 10251; Abu Dawud. 2350; classed as sahih by al-Albani in Sahih Abi Dawud. We will discuss its meaning according to the scholars below.)
Beginning and end of fasting in Ramadan
It is obligatory for the fasting person to refrain from things which break the fast from the true dawn until the sun sets.  What this means is when dawn breaks, not the adhan. Allah says (interpretation of the meaning): 
“and eat and drink until the white thread (light) of dawn appears to you distinct from the black thread (darkness of night).” [al-Baqarah 2:187]
So when a person becomes certain that the true dawn has come, he has to stop eating and drinking, and if there is food in his mouth he has to spit it out; if he does not do that, then he has invalidated his fast.
But if a person is not certain that dawn has come, he may eat until he is certain. The same applies if he knows that the muadhdhin gives the call before the time comes, or if he is not sure whether he gives the call on time or ahead of time: he may eat until he is certain. But it is better for him to stop eating as soon as he hears the adhan. 
Interpretation of hadith about eating during adhan in Ramadan
With regard to the hadith mentioned above, the scholars interpreted it as referring to a muadhdhin who gives the call to prayer before dawn breaks. 
Al-Nawawi (may Allah have mercy on him) said in al-Majmu’ 6/333: 
“We have stated that if dawn breaks and a person has food in his mouth, he should spit it out and complete his fast. If he swallows it after knowing that dawn has come, his fast is invalidated. There is no difference of scholarly opinion on this point. The evidence for that is the hadith of Ibn ‘Umar and ‘Aishah (may Allah be pleased with them both), according to which that Messenger of Allah (peace and blessings of Allah be upon him) said: “Bilal gives the call to prayer at night, so eat and drink until Ibn Umm Maktum gives the call to prayer.” Narrated by al-Bukhari and Muslim. There are many ahadith in al-Sahih with similar meanings. 
With regard to the hadith of Abu Hurayrah (may Allah be pleased with him), according to which the Prophet (peace and blessings of Allah be upon him) said: “If one of you hears the call (to prayer) and the vessel is still in his hand, let him not put it down until he has finished with it” – and according to another version he said: the muadhdhin used to give the call to prayer when dawn broke – al-Hakim Abu ‘Abd-Allah narrated the first version, and said: This is sahih according to the conditions of Muslim. Both were narrated by al-Bayhaqi, who said: If this is sahih, it is understood by the majority of scholars as meaning that the Prophet (peace and blessings of Allah be upon him) knew that he used to give the call to prayer before dawn came, thus the eating and drinking referred to were taking place just before dawn came.  
The phrase “when dawn broke” is understood to be the words of someone other than Abu Hurayrah, or it may be describing the second adhan, in which case the words of the Prophet (peace and blessings of Allah be upon him) – “If one of you hears the call (to prayer) and the vessel is still in his hand” – refer to the first call to prayer, so that it is in accordance with the hadith of Ibn ‘Umar and ‘Aishah (may Allah be pleased with them). He said: Thus the reports are in agreement. And Allah is the Source of strength, and Allah knows best.”
Ibn al-Qayyim (may Allah have mercy on him) stated in Tahdhib al-Sunan that some of the salaf followed the apparent meaning of the hadith mentioned in the question, and they regarded it as permissible to eat and drink after hearing the adhan of Fajr . Then he said:
“But the majority are of the view that sahoor should stop when dawn breaks . This is the view of the four imams, and the majority of fuqaha of the regions, and was also narrated from ‘Umar and Ibn ‘Abbas. The former quoted as evidence the words of the Prophet (peace and blessings of Allah be upon him), “… so eat and drink until Ibn Umm Maktum gives the call to prayer, for he does not give the call until dawn breaks.” 
It is narrated thus in al-Bukhari; in some versions it says: “… and he was a blind man who did not give the call to prayer until it was said to him: Dawn has come, dawn has come.” 
The majority quoted as evidence the words of Allah: “and eat and drink until the white thread (light) of dawn appears to you distinct from the black thread (darkness of night)” [al-Baqarah 2:187], and the words of the Prophet (peace and blessings of Allah be upon him), “Eat and drink until Ibn Umm Maktum gives the call to prayer,” and his words, “There are two dawns. 
The first one does not make it haram to eat or make it permissible to pray, but the second makes it haram to eat and permissible to pray.” Narrated by al-Bayhaqi in his Sunan.” 
Reports were narrated from some of the Salaf which indicate that it is permissible for a fasting person to eat until dawn breaks. Ibn Hazm (may Allah have mercy on him) narrated a large number of such reports, including the report that ‘Umar ibn al-Khattab used to say: “If two men are uncertain about dawn let them eat until they are certain.” 
Ibn 'Abbas said: “Allah has permitted drinking so long as you are uncertain, i.e., about the dawn.” 
Makhul said: I saw Ibn ‘Umar take a bucket of Zamzam water and say to two men, “Has dawn come?” One of them said: “It has come,” and the other said, “No (it hasn’t),” so Ibn ‘Umar drank (the water). 
Ibn Hazm said, commenting on the hadith asked about here and a number of similar reports: “All of this refers to cases when they are not yet certain that dawn has come. Thus the Sunnahs are in accordance with the Quran.” (Al-Muhalla, 4/367)
Undoubtedly most of the muadhdhins nowadays rely on clocks and timetables, not on sighting the dawn. This cannot be regarded as certainty that dawn has come. Whoever eats in this case, his fast is valid, because he was not certain that dawn had come, but it is better and safer to stop eating. 
Shaykh ‘Abd al-‘Aziz ibn Baz (may Allah have mercy on him) was asked: What is the ruling on the fast of one who hears the adhan of Fajr but continues eating and drinking? 
He replied: 
“What the believer must do is to abstain from things that break the fast, eating and drinking etc., when it becomes clear to him that dawn has broken, if the fast is an obligatory one such as in Ramadan and fasts in fulfillment of vows or as an act of expiation, because Allah says (interpretation of the meaning): 
“and eat and drink until the white thread (light) of dawn appears to you distinct from the black thread (darkness of night).” [al-Baqarah 2:187]
If he hears the adhan and knows that this is the call for Fajr prayer, he must stop eating. If the muadhdhin gives the call to prayer before dawn breaks, then he does not have to stop eating, and it is permissible for him to eat and drink until it is clear to him that dawn has come. 
If he doesn't know whether the muadhdhin gives the call to prayer before or after dawn, then it is better and safer for him to stop eating when he hears the adhan, but it does not matter if he drinks or eats something during the adhan because he does not know whether dawn has broken. 
It is well known that those who live in cities where there are electrical lights cannot know whether dawn has come at the time when it breaks, but they can follow the adhan and timetables which give the time of dawn, hour and minute, in accordance with the words of the Prophet (peace and blessings of Allah be upon him): “Leave that which makes you doubt for that which does not make you doubt.” And he (peace and blessings of Allah be upon him) said: “Whoever avoids doubtful matters will have protected his religious commitment and honour.”  And Allah is the Source of strength.” (Fatawa Ramadan, compiled by Ashraf ‘Abd al-Maqsud, p. 201)
Shaykh Ibn ‘Uthaymin (may Allah have mercy on him) was asked: You said that we must stop eating when we hear the adhan, but a few years ago they did not stop eating until the end of the adhan. What is the ruling on that action of theirs? 
He replied: 
“The adhan for Fajr prayer may be given either after dawn breaks or before. If it is given after dawn breaks, then one must stop eating as soon as he hears the call, because the Prophet (peace and blessings of Allah be upon him) said: “Bilal gives the call to prayer at night, so eat and drink until Ibn Umm Maktum gives the call to prayer, for he does not give the call until dawn has broken.” If you know that this muadhdhin does not give the call to prayer until dawn has broken, then stop as soon as you hear him. But if the muadhdhin gives the call based on a timetable he follows, or based on his watch, then the matter is less serious. 
Based on this, we say to this questioner: You do not have to make up what is past, because you were not certain whether you ate after dawn had broken. But in the future you should be careful; when you hear the muadhdhin, stop eating.” (Fatawa Ramadan, p. 204) 
And the Shaykh said, pointing out what is said about timetables being inaccurate: 
“Because some people nowadays are uncertain about the timetables that people have, and they say that they give the time of dawn too early, we went out into the desert with no lights around us, and we saw that the dawn came later, and some people exaggerated and suggested that the dawn comes twenty minutes later.  
But it seems that this exaggeration is not correct. What we think is that the timetables that people have nowadays give the time of dawn in particular as five minutes early, so if you eat whilst the muadhdhin is giving the adhan according to the timetable, it doesn’t matter. Some muadhdhins, may Allah reward them with good, give the adhan five minutes later than the time given in the timetables as they exist at present, as a precaution, but some ignorant muadhdhins give the adhan for Fajr early, claiming that this is on the safe side with regard to the fast, but they forget that they are being careless with regard to something that is more important than the fast, namely Fajr prayer. So people may end up praying ahead of time based on their adhan. If a person offers a prayer ahead of its time, even if he says the opening takbir, his prayer is not valid.” (Majmu’ Fatawa al-Shaykh Ibn ‘Uthaymin (may Allah have mercy on him), vol 19, question no. 772) 
And Allah knows best.