Until recently the term "alphabet" was used for any writing system where the symbols correspond approximately with phonemes, regardless whether both consonants and vowels are written as in all alphabets derived from or inspired by the Greek alphabet, or only the consonants are written, like in other writing systems derived from the old Semitic alphabet, without passing through the Greek alphabet.
Then the term "abjad" has been created, and also the term "abugida" (for alphabets where the base symbols are for consonants and the vowels are added as diacritic marks around the consonants), and the sense of "alphabet" has been restricted, in order to distinguish these 3 kinds of alphabets, but "alphabet" in the older wider sense can still be encountered frequently, either in the older literature or in informal speech, so one should be able to recognize both the stricter and the wider meanings.
In TFA, "alphabet" is used in the old wider sense. Moreover, it is not even used correctly in that sense, because they did not find a written "alphabet" like those used in teaching, but they have found a few written texts that are believed to have been written using an alphabetic script.
The oldest actual alphabets that have been found (which show the alphabetic order of the letters) are for the Ugaritic alphabet, which is older than the Phoenician alphabet, but much more recent than the oldest inscriptions that are believed to have been written with an alphabetic script.
Well, the actual scripts were distinguished semantically all along, and "alphabet" is also a word newer than the scripts in question. We should probably just use the words that make most sense to modern english speakers rather than... whomever you're referring to. Or just use "phonetic script" or something.
I almost get the sense that people are interpreting abjad as "lesser-than" an alphabet. It's just a distinction, it's not a value judgement.
Without additional conventions, "alphabet" would have been the appropriate name for any writing systems derived from the Phoenician alphabet, which include the majority of the writing systems based on alphabets, abjads or abugidas. The few other such writing systems, which have not passed through the Phoenician alphabet, are those derived from the Ancient South Arabian script, which for some reason had a different alphabetic order of the letters than the Northern Semitic alphabets, so it did not start with Alep and Bet.
You're doing a lot of equivocating between identity and ancestry. (e.g. saying that "alphabet" is composed of the names of two Phoenician letters rather than two Greek ones.)
In that framework, can we actually say that Ugaritic script is older than Phoenician script? My impression was that all of our knowledge of Ugaritic arises by coincidence, just that we happened to excavate a tell that turned out to keep records in what is as far as we know a script unique to it. Do we have reason to believe that someone wasn't using a proto-Phoenician script at the same time and we just haven't excavated them?
I just think it's useful to distinguish consonantal scripts from those with full vowel inclusion. Why not use alphabet/abjad for this? There's already a broad understanding of this meaning; why not lean into it?
I'll also admit this gets more complicated when I see people referring to an "abjad alphabet", but this leaves us with no way to describe an alphabet with consonants and vowels as opposed to a consonantal one.
For reference, "abjad" was introduced in 1990, so it's 35 years old. I don't think it's unreasonable to expect people to start using 35 year old terms to accurately describe academic subjects.