Alphabet is the correct word.
As I believe "alphabet" refers specifically to the latin a-z
Your assumption is wrong.
The etymological origin of the word "alphabet" relates to the Greek alphabet, in which the first two characters are "alpha" and "beta".
However, "alphabet" is currently used for any comprehensive set of characters. Take note of definition 1.1, which shows you that it can be used more loosely and does not even need to refer to a linguistic character set.
A set of letters or symbols in a fixed order used to represent the basic set of speech sounds of a language, especially the set of letters from A to Z.
‘the first letter of the alphabet’
‘a phonetic alphabet’
1.1. The basic elements in a system which combine to form complex entities.
‘DNA's 4-letter alphabet’
The OED does mention that it can be taken to refer to the A-Z alphabet, but not exclusively.
You've somewhat answered your own question. From your question body:
English, French, German etc are all languages, but all use the Latin alphabet.
If you assume that "alphabet" inherently refers to the Latin A-Z, then why didn't you say:
English, French, German etc are all languages, but all use the alphabet.
The fact that you have to say "the Latin alphabet" contradicts your notion that "alphabet" always refers to the Latin A-Z. That would make "Latin alphabet" a pleonasm.
Needing to specify which alphabet means that there is more than one alphabet.