The term "syllabary" is used in a lot of ways (the term is hundred of years old, so that's no surprise). So it partially depends on what your criteria for calling something a "syllabary" are. If we take a syllabary to refer to a language where each primitive graphic unit represents exactly a syllable, then there are not many syllabaries. Linear B is not a syllabary, for example, because "ai-ku-pi-ti-jo" uses a "syllable" grapheme (pi) to represent a part of the syllable – there is no grapheme for the entire syllable gup. Cherokee comes close, because most syllables in the language are open; but, in fact, there is a grapheme for coda s, which is not a whole syllable (the other problem is that Cherokee simply glosses over some distinction such as ka / ga; vowel length is ignored as is coda h). Japanese katakana likewise doesn't actually have a system where each grapheme is a complete syllable (the single syllable kjoo is represented as きょう with three graphemes, and so on). Cree and Ojibwe writing are similar, with the added quirk that the graphic primitive could be deemed to be the Ca shape with orientation and the length diacritic overriding intrinsic [a]. Plus, as with other supposed syllabaries, coda consonant are in fact separate letters. Devanagari is in some ways similar to Ojibwe, in that the "primary" graphic elements represent Ca (with add-ons to indicate "no vowel" or "another vowel"). Such systems are in fact called "abugidas".
The broadest notion of syllabary refers to something like "a writing system where the syllable is a basic organizational unit", and there are very many languages like that. When the initial consonant of the syllable defines the fundamental shape and vowel choice is indicated by a diacritic-like addition, we call that an "abugida", also "alphasyllabary". In other words, the terminology is not particularly precise or subtle, and therefore there aren't a lot of terms for referring to different distinctions that could be made.