Do all syllables belong to open syllable or closed syllable?

Is there any other category for a syllable which is neither an open nor closed syllable?

English has a rather unusual syllable structure that allows for as many as three leading and five trailing consonants. As a result, there is arguably some ambiguity as to where the articulation point between two syllables falls. However, there is not universal agreement about this point. Wikipedia offers this reasonably clear analysis:

In English, consonants have been analyzed as acting simultaneously as the coda of one syllable and the onset of the following syllable, as in 'bellow' bel-low, a phenomenon known as ambisyllabicity. It is argued that words such as arrow /ˈæroʊ/ can't be divided into separately pronounceable syllables: neither /æ/ nor /ær/ is a possible independent syllable, and likewise with the other short vowels /ɛ ɪ ɒ ʌ ʊ/. However, Wells (1990) argues against ambisyllabicity in English, positing that consonants and consonant clusters are codas when after a stressed syllable followed by an unstressed syllable, or after a full vowel and followed by a reduced syllable, and are onsets in other contexts.

As far as I can tell, the term 'ambisyllabic' applies specifically to consonants, not to syllables; that is, one could speak of an "ambisyllabic consonant" that seems to belong to two different syllables, but not of an "ambisyllabic syllable." However, this article uses the term 'half-closed' to describe syllables that contain ambisyllabic consonants, and that seems like as good a term as any.

There are actually names for syllables not included in the open and closed syllable categories:

Apparently, there are six types of syllables:

There are six types of syllables:

A closed syllable ends in a consonant. The vowel has a short vowel sound, as in the word bat.

An open syllable ends in a vowel. The vowel has a long vowel sound, as in the first syllable of apron.

A vowel-consonant-e syllable is typically found at the end of a word. The final e is silent and makes the next vowel before it long, as in the word name.

A vowel team syllable has two vowels next to each other that together say a new sound, as in the word south.

A consonant-le syllable is found in words like handle, puzzle, and middle.

An r-controlled syllable contains a vowel followed by the letter r. The r controls the vowel and changes the way it is pronounced, as in the word car.

This is backed up by this site:

Six Kinds of Syllables

There are six different kinds of syllables in English: