> This is a modern distinction.
Well, the actual scripts were distinguished semantically all along, and "alphabet" is also a word newer than the scripts in question. We should probably just use the words that make most sense to modern english speakers rather than... whomever you're referring to. Or just use "phonetic script" or something.
The word "alphabet" has been used at least since the second century AD (e.g. by Tertullian), but it is composed from the names of the first 2 letters of the Phoenician alphabet, names that must be at least 3 millennia old.
Without additional conventions, "alphabet" would have been the appropriate name for any writing systems derived from the Phoenician alphabet, which include the majority of the writing systems based on alphabets, abjads or abugidas. The few other such writing systems, which have not passed through the Phoenician alphabet, are those derived from the Ancient South Arabian script, which for some reason had a different alphabetic order of the letters than the Northern Semitic alphabets, so it did not start with Alep and Bet.