Praise be to Allah, the Lord of the Worlds; and blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.
First of all I'd like to tell you that there is a difference between Qur'anic script (Rassem) and the different styles of writing in the Arabic language Ruqaa', Naskh, Kufi …. Etc.
The Qur'anic Rassem is how the letters and words of the Qur'an are transcripted in the Moushaf (Holy Qur'an Book). Qur'an now is written in the same way it was written in the Moushaf which Caliph Othman Ibn Affan ordered some Companions headed by Zaid Ibn Thabit to record and he (Othman) dispatched those Moushafs to all Muslim countries at that time.
The way in which the Qur'an is written now is called Rassem and it is peculiar to the Qur'an. Rassem originates from the dictation of the Prophet of the Qur'anic verses. The Prophet used to dictate the Qur'an as it is revealed to him to the "Wahy Writers"; the writers of the Qur'an who were a chosen group of the Companions. These writers would write the Qur'an the way they knew under the supervision of Prophet Muhammad (Peace and blessing be upon him).
Qur'an must be written according to Rassem rules. So it should not be written according to modern Arabic linguistic rules.
So Rassem should be preserved as its protection is part of the maintenance of the seven letters (i.e., the seven dialects of the Arabic language when the Qur'an was revealed i.e. Quraish, Hudhail, Tameem ……) in which the Qur'an was revealed.
The Muqrie (reader of the Qur'an) is a scholar in the different forms of reciting Qur'an as reported orally from one scholar to another till the authentic chain reaches the Prophet (Peace and blessing be upon him).
There are seven different recitations and they are not the same as the seven different letters (dialects). Each one of these seven recitations is called after an Imam who was known to have mastered it. These Imams are: Nafi, Assim, Ibn Katheer, Al Kassai, Ibn Amer, Abu Amr Ibn Alaa and Hamza.
As for the names you mentioned they are just names of some famous contemporary reciters of the Qur'an who are numerous.
Allah knows best.