Which is the correct terminology, istiaazah or ta'aawwudh, and basmalah or bismillah?

Question

Dear bothers and sisters @ abouttajweed.com, I pray all is well and in good Islamic spirits IA.

First of all I must thank you and say Jazakallah-hu khairun for your absolute brilliant work, May Allah Azzawajal grant you all the highest place in Jannah and give Barakah in your lives and Guide and protect us and take us from this world in the state that he the Almighty, is please with us. Aameen!

 

Answer
Wa alaikum assalaam wa rahmatullahi wa barakatuh. 
Jazakum Allahu khairan for your kind words and ameen to your dua'.  May Allah give you the same, insha' Allah.

1.  means seeking refuge from Allah, usually in the words: .   Both terms and Ta'awudh serve the same purpose and mean the same thing. 

2.  is an abbreviated term used for the statement or simply bismillah. Bismillah is really  "bi ism Allah" which means, I start in the name of Allah.  So the words we say are "bismillah", or more completely:: , and the term we use when referring to these words, but not saying them specifically is the Basmalah. 

Please see: http://www.abouttajweed.com/seeking-refuge-and-the-basmalah-in-recitation/seeking-refuge-and-saying-the-basm.html

Wa alaikum assalaam wa rahmatullah.