It should also be noted that a difference is often made between alphabets in the strict sense, where consonants and also vowels are represented by distinct symbols, and alphabets in the wider sense, where this is not the case (vowels are not represented at all or occasionally by certain consonant symbols typically when clarification is necessary). A writing system where symbols denote larger units of speech is not called an alphabet, but a syllabary. If it does not represent phonetic, but semantic units, it is called a logographic script. There are of course all kinds of mixed forms ("I ♥ NY").
Technically, a syllabary only refers to writing systems where the symbol represents the specific consonant and vowel pair, such as Japanese's Hiragana. For example, in a syllabary, the syllables "ka" and "ki" are two different symbols.
If the vowels are optional or not present, e.g. there's one "k" symbol regardless of the vowel, it's an Abjad. The archetypal Abjad is the Hebrew writing system.
If the vowels are written by adding them to the consonant symbol (similar to diacritics), it's called an Abugida. One example of this is the Ge'ez script in Ethiopia.
I did not want to make it too technical, so "Abjad" falls under "alphabets in the wider sense" and "Abugida" under "mixed forms". My comment was based on the assumption that the article in question does not necessarily refer to an alphabet in the strict sense. To make this clear, I did not think it was necessary to go into too much detail.
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.