They were called articles by analogy to other languages. Determiners is too broad a class, encompassing too many different words with clearly different roles. It happens that in English articles and other determiners such as possessives or demonstratives are exclusive, so this is harder to see.
But in other languages, a noun can have both an article and a possessive, or both an article and a demonstrative. For example, in Italian, Portuguese, and Romanian, you don't say "my dog", you say "il mio cane" / "o meu cão" / "câine-le* meu". In Romanian you even have to say "câine-le* ăla" for "that dog".
English also makes it hard to tell different word classes apart because it has very little variance in words. In other languages, articles have to agree with their nouns in different ways from possessives and/or demonstratives. For example, in German the possessive pronoun has to agree with both the gender of the noun it describes and that of the noun it replaces - "his dog, her dog, his cat, her cat" become "sein Hund, ihr Hund, seine Katze, ihre Katze", whereas of course the definite article only agrees with the gender of the noun it attaches to. Demonstratives in German don't agree with the noun in gender at all.
Also, quantifiers like "some", "none", "a few" etc are different from the above in that they can be used in more ways than just determiners. They can appear standalone as the subject of a phrase, for example, which articles and possessives can't.
* note that standard spelling is "câinele", which is "câine" + the definite article "-le", since definite articles become noun suffixes in Romanian.