In as many times as we have been on the internet or in a discussion or a chat group, we almost always run into someone who feels offended that we use the word "God" when we talk about the One and Only God (Allah in Arabic).
It is surprising to notice that most of these Muslim brothers and/or sisters do not know that the word "Allah" is the Arabic word for the word "God". Many of them believe that "Allah" is the name of the Muslim God. They do not realize that the word "Allah" does not belong exclusively to the Muslims and that it has always been used before (and after) Islam by the Arabic-speaking Jews and Christians when they speak about God.
Talking to English speaking people about God using the word "Allah" is very much the same like speaking to Arabic speaking people about "Allah" using the word God. It makes all the sense to show respect to the people and their language by speaking to them in the language they use. Insisting on the use of the word "Allah" which is the Arabic word for God immediately creates the illusion that "Allah" is a whole different deity than God of the whole world. It creates a god that belongs ONLY to the Muslims, and takes the universality of Islam out of it.
Although a lot has been said about the philology of the word ‘Allah’, however, in my opinion, the former of the two opinions noted by you seems to be closer to the correct one. A detailed discussion compiling the opinions of various scholars of the Arabic language regarding the origin of the word can be seen in “Lisaan al-Arab” under the word “Aliha” (a-l-h). In my opinion, ‘Allah’ is an Arabic word meaning ‘the God’. According to the general principle of making proper nouns from common nouns in the Arabic language, the word “ilah” (common noun) has been converted to “al-ilah”, which became “Allah” due to the turgidity and the slight difficulty of pronunciation of the word “al-ilah”.
The Quran, because its prime and first addressees were the Arabs, used the word “Allah” for the Supreme Being, as that had traditionally been the word used for the Supreme Being in that language. The same had been the case in the older scriptures. Those scriptures, like the Quran, used those words for the Supreme Being, which had already in vogue in those languages, to refer to the Supreme Being.
However, there have been scholars of the Arabic language who ascribe to the opinion that “Allah” is the actual name of the Supreme Being. It is indeed important to the evidence that they have provided into account. Nevertheless, I feel that to give God a name is a requirement of us, humans. God, being the absolute being is in no need for a name.
May the Almighty guide us all to the path of His liking.