You have answered your own question, but (食べ)たい, (食べ)ない, (食べ)た, (食べ)ます and so on are almost always called 助動詞 in traditional 国文法 studied by 国語学者/日本語学者* and taught by Japanese middle/high school teachers. These are always called 助動詞 in monolingual dictionaries, too. Importantly, these are words on their own, and not part of a verb's conjugation form.
(* I personally hesitate to call them 言語学者 for a reason, but I won't go into the details about it...)
For example, 書かない consists of two words, 書か (未然形 of 書く) + ない (助動詞 for negation). 書きたい is also two words, 書き (連用形 of 書く) + たい (助動詞 for desire). 書いた is 書い (another 連用形 of 書く) + た (助動詞 for past tense). 書きたくなかった has all the four words, 書き (連用形 of 書く) + たく (未然形 of たい) + なかっ (連用形 of ない) + た (終止形 of た).
This is different from what you may have learned as a JSL (Japanese-as-a-second-language) learner, but that's how the Japanese grammar is usually taught and discussed in Japanese.
It would be possible that they are directly referred as 活用形 katsuyōkei.
Indeed, if you're seeing only simple examples like 歩いた, it may be hard to see how it's different from "walked". The main reason these 助動詞 are considered independent words is that they can be combined to say complex things.
For example, 食べさせられたくなかった means "did-not-want-to-be-made-to-eat", but it's odd to call something like this a "form" when it can be logically split into smaller parts. According to 国文法, this is a "simple" combination of 6 words, 5 of which are 助動詞: 食べ + させ (causation) + られ (passive) + たく (desire) + なかっ (negation) + た (past).
書きませんでした is often introduced as a "form" (negative-polite-past-form) in JSL textbooks, but this is just for the sake of convenience. The rules governing 助動詞 are a little too tricky for a beginner, so it's faster to learn them by rote as a form at first.
For another thing, while there are a few irregular verbs in Japanese, their irregularity does not cross word boundaries and affect the following 助動詞. For example, 行く is a well-known irregular verb whose "past form" is not 行いた but 行った, but you can see the た part remains た. This is another reason why 助動詞 are considered separate. In English, the past form of go is went, and it's impossible to see two words in it.
I still wonder if they are not classified as other parts of speech (for instance 助詞 joshi), or if they are considered suffixes (接尾辞 setsubiji).
助動詞 are clearly different from 助詞 (particles; を, が, へ, から, etc):
- 助詞 usually follow 体言 (nouns), whereas 助動詞 usually follow 用言 (verbs and i-adjectives).
- 助詞 never conjugate, whereas 助動詞 can conjugate by themselves. For example, たい and ない themselves conjugate like an ordinary i-adjective (たく, たかっ, たけれ; なく, なかっ, なけれ), allowing another 助動詞 to follow it.
- In terms of purpose, 助詞 is closer to English prepositions (to, of, at, etc) while 助動詞 is closer to English auxiliary verbs (can, should, must, etc).
A 接尾辞 (suffix) forms another single word. For example, in English, "boys" is still one word, and "brighten" is still one word. However, a Japanese 助動詞 is a bit too flexible to be called a part of a word, as described above, and that's why they are not called suffixes.
Related:
(EDIT: Oh, I noticed 接尾辞 in Japanese Wikipedia does not agree with what I explained above. So this is one way where 国文法 and 言語学 differ... Whoever wrote this seems to think even が and は are suffixes based on some linguistic theory, but I don't think those who adhere to 国文法 would agree.)