Praise be to Allah.
The times of prayer have been clearly explained in the religious texts, and they are connected to observable matters; anyone can work it out with a little thought and reflection.
The time for Fajr begins with the break of the true dawn, in which light extends along the horizon, right and left.
The time for Zuhr begins when the sun passes the zenith, i.e., when it passes the middle of the sky. That may be known by the lengthening of shadows, after the shadows reached their shortest length.
The time for ‘Asr begins when the shadow of a thing is equal to its length, after the sun has passed its zenith.
The time for Maghrib begins when the disc of the sun has completely disappeared.
An-Nawawi said: What this means is that the sun’s disk has dropped completely below the horizon. After it has completely disappeared, it does not matter if some of its rays are still there; rather the time for the prayer begins even if the rays are still present.
End quote from al-Majmoo‘ (3/33).
The time for ‘Isha’ begins when the red afterglow on the horizon disappears.
For more information on the times of prayer, please see the answer to question no. 9940.
Whoever does not know the times of prayer should follow someone who does know that. That includes following timetables.
If the timetables differ, then he should err on the side of caution.
The Maaliki scholar Shaykh Shihaab ad-Deen al-Makki (may Allah have mercy on him) said in Irshaad as-Saalik (1/13): If someone is uncertain as to whether the time for prayer has begun, he should not pray; rather he should delay it until he is certain or thinks it most likely that the time has begun. End quote.
You should err on the side of caution with regard to the fast by starting to fast at the earlier time given [in the timetables], and you should err on the side of caution with regard to Fajr prayer by praying at the later time. Thus you will be certain that your fast and your prayer are valid.
In order to err on the side of caution with regard to Maghrib, you should break your fast and pray at the later time given.
You could go out to an area of flat land and see the sun set and its disk disappear completely, in order to verify the timetables that you have, because working out the time of Maghrib is easy, unlike Fajr.
There is nothing wrong with using two times for Fajr, one for starting the fast and the other for praying, so as to be on the safe side with regard to two acts of worship, for one who is not sure whether the time given is correct.
And Allah knows best.