Related (as they were influenced by the Cherokee writing system) and similarly interesting: Canadian aboriginal syllabics [1], which are unique in that the shape of the character determines the consonant, while the 90-degree rotation of the character determines the vowel.
Strictly speaking, the Canadian Aboriginal syllabics is an abugida, not a syllabary.
In the categorization of writing systems, alphabets (such as Latin) encode each phoneme as a distinct letter; abjads (such as Arabic) encode only consonants, with vowels as optional diacritics; abugidas (such as Indic scripts) primarily encode consonants but mark vowels in some systematic way; syllabaries just encode each syllable as a unique glyph without any systematic modification; and logographic systems don't have any systematic phonetic component (although they usually are influenced by phonetics, e.g., the rebus principle of diagramming abstract concepts by use of homonyms).
Sequoyah had access to an English-language Bible, so although he couldn't read the Latin text, he used several letters as inspiration for Cherokee, which is why you see Latin characters that correspond to completely different sounds.
While I agree the distinction you reference between abugidas and syllabaries is useful, it's not a distinction that's always observed (e.g. Indic scripts are sometimes referred to as syllabaries).
[1]: https://en.wikipedia.org/wiki/Canadian_Aboriginal_syllabics