Would abu al-Hassan al-Asha'ari regard himself as among the Asha'arites?

The Ash'arite أشعرية‎ or Asha'irah ٱلْأَشَاعِرَة are a known theoligcal sect of creed that refers by name to abu al-Hassan al-Asha'ary and most of its followers can be found among followers of the shafi'i and maliki school of fiqh. And they regard themselves as representatives of ahl-as-Sunnah wa Jma'ah beside other sects (of creed) such as the Maturidi's and the Salafi's. Now I've read somewhere -correct me if I'm wrong- that abu al-Hassan al-Asha'ari passed through three phases in his life. 1. A phase in which he was a follower of the creed of al-Mu'tazilah . 2. A phase in which he started debating al-Mu'tazilah based on 'ilm al-Kalam . 3. A phase in which he defended the creed of ahl as-Sunnah wal Jama'a. And as far as I know in each of these phases or at least in the last two he has compiled books defining his creed and theology. My question now is: Those scholars who claim to be 'Ashra'arites like the imams and scholars: - Among the Shafi'is: al-Juwayni, al-Ghazali, al-Bayhaqi, an-Nawawi, as-Suyuti, al-Fakhr a-Razi, ibn 'Abdassalaam, as-Subki, ibn 'Asakir ... - Among the Maliki's: al-Baqilani, abu Bakr ibn al-'Arabi - Among the Hanbali's: ibn al-Jawzi (at least wikipedia claims he was because of his works in ilm al-Kalam) - ... which of his creeds or phases did they follow? It could be great if your answer could include some details.