What are the two names of Allah, which He chose to represent Himself by?

The verse in the Quran in which Allah chooses two names to represent Himself is,

“Call on Allah or call on The Most Merciful…”[1]

Of these 2 names, the first mention is of the name ‘Allah’. If these two names, Allah and Ar Rahman, are the greatest of the names, then naturally the best name that a person could call himself is a servant of Allah or the servant of The Most Merciful. Even when Prophet Muhammad (ﷺ) mentioned these two names, he mentioned Abdul Allah first, and then he mentioned Abdul ar Rahman. And if we look at the two names in any case, whether in the verse, in Surah Al Isra, or in this hadith, we should note that ar Rahman addresses the issues of mercy. Fundamentally, it is a derivative of rahmah or mercy; it is the essential meaning of the name Ar Rahman. It may be mercy in its comprehensive sense, that it encompasses all of His creation, and there are other aspects of His mercy. But it essentially is mercy. Whereas, the attribute of might and power, it is not represented in Ar Rahman. However, it is represented in the name Allah. The might and the power of Allah, al Qaadir and al Aziz; all are found in the name Allah, just as mercy and magnificence, all of these are found in the name Allah.

References:

[1] Quran 17 : 110