Wa alaikum assalaam wa rahmatu Allahi wa barakatuh.
Imam Ibn Al-Jazaree researched many sources and outlined different authentic ways in his book An-Nash fi-l-qiraa’aati-l-3ashr. The poem he wrote to make learning these ways easy is called At-Tayyibah. The ways then are called tareeq at-Tayyibah. There are around 50 ways outlined for Hafs ‘An ‘Aasim in tareeq at-Tayyibah. One of these ways is tareeq Ash-Shatibiyyah. Tareeq Ash-Shaatibiyyah gets its name from the poem written by Imam Ash-Shatibiyy on the seven qira’aat, that is popularly known as Ash-Shatibiyyah; this is the way the vast majority of Muslims recite this day. .
The differences in the ways are sometimes as simple as to whether , for example is read with a sakt or not, or whether is stopped on with an alif or not, or both ways are allowed, etc. These words are the words found in the lessons on the special words for Hafs ‘an ‘Aasim by the way of Shaatibiyyah, not all the ways of Hafs read them the way they are read in tareeq Ash-Shaatibiyyah. Some rules in some ways change from one way to another, for example: there are some ways that have a ghunnah with the idghaam of the saakinah and tanween with the letters and . Some other ways have a breathless pause on a saakin letter before a hamzah. There are differences in the lengths of both and throughout the various ways. Some ways have six vowel counts for , others have five. There are some ways have three vowel counts for , others two, and still others five. There are some ways that have takbeer before every surah of the Qur’an, not just before or after the surahs at the end of the Qur’an.
Each way though, has its own distinct rules and must be followed completely. One cannot take something from one way and mix it with another way. Each way has an authentic chain of transmission from the Prophet, to the reciters who have ijaazah in these ways this day.
Insha’ Allah this gives you some information. We want to stress too that one should learn and master Hafs ‘an ‘Aasmin by the way of Ash-Shatibiyyah before going on to learn the various ways of Tareeq At-Tayyibah.
May Allah reward you for your good questions and increase you in beneficial knowledge.
Wa assalaam alaikum wa rahmatu Allahi wa barakatuh.