It says that it's the middle part of a vowel.

No the whole article doesn't use the word "middle" even once if you do a quick search.

You don't put a w or a y in the middle of a syllable, but it's frequent to put the corresponding sound there.

You are still thinking in terms of letter instead of phonemes.

Your definition is not the one in that article. If you look at the vowel chart they all follow a similar schema.

But there also a definition on wikipedia:

In phonetics, a vowel is a sound in spoken language, such as an English ah! [ɑː] or oh! [oʊ], pronounced with an open vocal tract so that there is no build-up of air pressure at any point above the glottis. This contrasts with consonants, such as English sh! [ʃː], there is a constriction or closure at some point along the vocal tract.

For example, bird, herd, and turd just have the r sound.

In those examples "ir", "er" and "ur" are together the vowel "əː". There no real "r" sound in those words. In contrast words like "rare" or "sorry" actually have the "r" phoneme.

"əː" is a vowel and "r" is a consonant. To be more precise "r" is the voiced alveolar fricative while "ə" is the mid-central vowel also known as the schwa and the ":" suggests that it's long.

"ə" appears even two times in violet without there being any letter "r" in the word.