Does Allah mean God?
Yes (if one intends to refer to "Allah") and no (because it strictly means more than "God" expresses).
Allah الله is a word that describes a deity and its uniqueness in one word.
God or deity in Arabic means Ilaah it is written
إلاه or with an article: الإلاه (al-Ilaah)
إله (In the qur'an often written with a pending letter "alif" between the "laam" and "haa" (last two letters). Or written with an article: الإله
Usually to make this God or deity determined and ideally unique an article -in Arabic "ال"- is required.
This usually is sufficient enough with almost all of Allah's names: such as ar-Rahmaan الرحمن vs. Rahmaan رحمن, ar-Raheem الرحيم vs. Raheem رحيم, al-Baasit الباسط vs. Baasit
Can a human be given the name Barr or Al-Barr?
Tawheed asma was siffat
But in case of the word Allah some specification of the Arabic language apply:
As you may read above "the deity" is "al-Ilaah", but this is not determined enough to show and fix the issue of the uniqueness therefore the word Allah is more expressive, accurate and explicit to express the uniqueness.
Therefore the shahadah or part of the adh'an is:
"La ilaaha illa Allah"
There's no God (deity), but Allah
لا اله الا الله
If God was 100% equal to Allah one couldn't differentiate between Ilaah and Allah in this sentence, but this is not the case.