I have been getting back into my grapholinguistics/grammatology/graphonomy study again recently. Comparing different readings so far has been really interesting, especially when it comes to typology of the world's scripts. When I studied grapholinguistics as a unit at uni, we were introduced to ultimately 6 kinds of scripts:
Logography/Morphography - Each grapheme represents a word/morpheme in a language
Syllabary - Each grapheme represents a syllable in a language, similar syllables, be it in their rhyme or onset, aren't necessarily graphically related (Cherokee syllabary, Japanese Hirigana and Katakana)
Alphasyllabary - As above, but there is some system to how the syllables are constructed. Many alphasyllabaries that I can call to mind tend to have a base which represents their consonant onset, and then some modification for rhymes, such as a mark like an akshara (in the case of Indic scripts), or rotations (in the case Cree Syllabary and Qaniujaaqpait)
Abugida - A script which systematically does not write a vowel and is left implied, typically its most common vowel sound is the one unwritten (like the a sound in Hindi, with Devanagari. How the base syllable is nessecarily Ca, and adding vowel markings or (as my prof would call them) vowel killers, result in different CV and C readings)
Abjad - Each grapheme represents consonants, and does not write any of its vowels. These typically belong with Semitic languages whose grammar and morphology fills in the vowels, and so native speakers would be able to infer the vowels based on context. Modern abjads tho do also have vowel markings, but these are largely optional, and from what I have heard are used in helping people read old texts, and learners of these languages that use these scripts
Alphabet - Each grapheme represents either consonants or vowels
While this does make sense, abugida's definition feels like it is a way of describing how a writing system deals with a specific sound, and doesn't really help provide a general idea of how a script works, as any script with a way of representing vowels expressly, like alphabets or alphasyllabaries, can then have the ability to not one of their vowels and leave it to inference.
In my readings I have done reccently so far, it is interesting to see how this typology has changed, first starting with logographies and phonographies, moving onto logographs, syllabaries, and alphabets, then moving onto logographies/morphographies, abugidas (which Peter Deniels, in SECTION 1 of "The World's Writing Systems", describes with the definition I have learnt for alphasyllabaries), abjads, and alphabets. This expanding in classification shows a growing field of study gaining more and more knowledge as time goes on as we consider more and more writing systems! They are so so cool, and I feel that writing systems can tell us so much about a language! It can tell us its history, how people view their language, the way writing is used, and can be used to express linguistic and paralinguistic ideas is fascinating! And the choices we make with scripts and writing can reflect social realities! Different language perceptions, different style perceptions, hell even different script perceptions! It is all so cool and fascinating and even tho I am currently not actively studying this field for my undergrads, I hope I can do more with it in the future!