Probably most of you do not know that we are all using a variant of the ancient Phoenician alphabet. Actually I am not sure if that is precisely true, as I think the Phoenician alphabet was preceded by an Assyrian one. But at any rate, our classic Western alphabets all came out of the Levant and Mesopotamia in some way or other. Indeed, it is even theorized that many of the syllabaries in use in Central, South and Southeast Asia are also rooted in this original alphabet from the Levant.
Of course, Chinese and consequently Korean and Japanese alphabets have another origin.
One might wish to throw the odd SE Asian orthographies such as Thai, Lao, Burmese, Vietnamese, Javanese, Sundanese and Khmer there, but my understanding is that all of those SE Asian orthographies were actually derived from syllabaries originally designed in India.
A few writing systems such as Georgian, Armenian and Cree may have been created de novo, but I might have to look that up. The only non-Middle Eastern derived orthography that immediately comes to my mind is the Chinese ideographs.
The origins of the Assyrian/Phoenician alphabet appear to have been ultimately in Egyptian hieroglyphics. So the ancient Egyptians really started it all when it comes to writing down words, at least for the West.
Chinese ideographs may date from even earlier. Chinese bone writing goes way back.
Very early European writing such as runic systems and similar systems in Asia such as the Turkic Orkhon inscriptions may not be related to the Phoenician system at all. The Yukaghir in Siberia and the Yi in South China may also have designed de novo systems.
I’m willing to be corrected, but my impression was that while the Phoenicians (Lebanese) developed the first recorded alphabet, and they gave the Greeks the idea, the Greeks took it and did something very different. For example, they added vowels.
English speakers use the Roman or Latin alphabet, which was definitely derived from the Greek, but its different enough that they can’t read anything written in the Greek alphabet without special training.
The ancient Assyrians used Cuneiform, which I do not believe is directly related to the Phoenician alphabet. All the alphabets in use in Europe and the Middle East are derived from the Phoenician by various routes. Arabic and Hebrew are derived from the Aramaic alphabet, which lineally descends from Phoenician. The Latin and Cyrillic alphabets are adaptations of Greek, which was a direct borrowing of Phoenician.
Origins of the Brahmi script (the ancestor of India’s modern Devangari alphabet) are obscure, but there is some reason to believe it comes from West Semitic (i.e. Phoenician) sources. So your first statement is correct.