There are many specialized terms for different types of writing system, but those distinctions are generally of very little interest unless you're compiling a table of different writing systems.

Generally you look at what concepts are embodied in the script, and at the form of the glyphs. So:

You might have a script that assigns glyphs to phonemes. ("Language is made of sounds.")

You might have a script that assigns glyphs to consonants and doesn't bother to represent vowels. ("Language is made of sounds, and some of them are more important than others.")

You might have a script that assigns glyphs to syllables. ("Language is made of things you can say.")

You might have a script in which the glyphs assigned to syllables are composed of recognizable and conceptually distinct parts, but those parts have no independent representation. (Compare the glyphs ሀ ለ ሐ with the related glyphs ሄ ሌ ሔ.) ("Language is made of things you can say, but there are patterns.")

You might have a script that assigns glyphs to words, though in almost all cases you don't. The label "logographic script", applied to a script the labeler doesn't know well, is infinitely more popular than the concept "logographic script". I don't think any script has ever existed meeting the criterion of "it does not represent phonetic, but semantic units". But there are some, and used to be more, that leaned more or less strongly in that direction.