All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and messenger.
Prayer occupies a great status in Islam that is not shared by any other act of worship. Prayer is one of the indispensable and essential “pillars” of Islam. The Prophet, sallallaahu ‘alayhi wa sallam, said: "The head of the matter is Islam, its pillar is prayer and its highest peak is Jihaad in the Path of Allaah."
Add to this, prayer is the only act of worship that was Prescribed directly from Allaah The Exalted to the Prophet, sallallaahu ‘alayhi wa sallam, without any mediation from above the seventh heaven. This denotes the great status that prayer has in Islam. Anas ibn Maalik reported that the Prophet, sallallaahu ‘alayhi wa sallam, said: "On the night of my Ascent to the Heaven, fifty obligatory prayers were prescribed, then the obligation was reduced to five. Then, a caller proclaimed, 'O Muhammad! These are five prayers and they are all (equal to) fifty (in reward) for My Word does not change.'" [At-Tirmithi and Ahmad]
Muslims have unanimously agreed on the obligation of prayer. Allaah The Exalted Says (what means): {… establish prayer and give Zakah …}[Quran 73:20]
Muslims also have unanimously agreed that the obligatory prayers in Islam are five prayers in a day. When the Prophet, sallallaahu ‘alayhi wa sallam, sent Mu‘aath to Yemen, he said to him: "… Inform them that Allaah Has Enjoined five prayers upon them to be offered a day …" [Al-Bukhari and Muslim]
Based on this narration and the Muslim consensus, it should be clear for the dear questioner that there are only five obligatory prayers in a day in Islam. As for the six prayer times in the agendas and the calendars, these are not the obligatory prayers in Islam. They add the sunrise time just to identify the due time for performing the Shurooq prayer (i.e. supererogatory prayer after sunrise) and nothing more. This is to provide more information and to define the time when offering supererogatory prayers after Fajr is lawful.
As for your question about the difference between the Fajr and morning prayers, the answer is that both Fajr and morning are synonymous names for the same prayer. There is a two-Rak‘ah voluntary prayer before the Fajr prayer that is called the Sunnah for the Fajr or morning prayer.