Narrow Transcription: How is "very narrow allophonic" transcription identical to "impressionistic" transcription enabling something extra?

Question

English: "check the lens well"

1. Phonetic (broad) transcription: /tʃɛkðəlɛnzwɛl/
2. Impressionistic narrow transcription: [tʃe̞ʔ͡kð̞əlɛ̆nzwæ̠ɫ]
3. (One) very narrow allophonic transcription: [tʃe̞ʔ͡kð̞əlɛ̆nzwæ̠ɫ]

I understand what allophones are, but when the IPA handbook mentions:

The difference [b/w the impressionistic transcription and the allophonic transcription] is that NOW [with the latter] it would be possible to express, in conventions, the relation between the allophones transcribed and the phonemes which they realize...

...what does the italicized line mean? I mean as #2 and #3 above are identical and we didn't add any new information b/w the two of them; how is then only the latter of the two enabling in conventions, the relation between the allophones transcribed and the phonemes which they realize; while the former couldn't. Thanks!

Answer 1

Transcriptions of type 2 aim to symbolically record the physical event, and do not have an language-specific interpretation. They are, essentially, "what you hear", and when you first encounter a language and record utterances in the language, you are simply transducing the acoustic waveform into fixed letters. This is the foundation for transcriptions of type 3, where you analyze the grammatical distribution of sounds in a language and devise a set of allophone-to-phoneme relations. The sequence of letters might be exactly the same, though usually they are not, but they do not delve into how that is possible (that's out of the scope of that work). In fact, most transcriptions tend towards type 1 or weakly type 3. From the speaker-oriented perspective, type 2 is the body output, and type 3 is towards the end of the phonological output (hence the parenthesized "one").

The difference resides in the ontological interpretation of the symbols, since IPA is neutral as to whether it represent listener impressions, body outputs, grammar outputs or intermediate representations in the linguistic processing system. Type 2 is maximally free of grammatical and linguistic interpretation.

Answer 2

It is only the allophonic transcription of that narrowness you chose to give as an example in your question (#3) that totally coincides with the impressionistic narrow transcription (#2). If you choose a lesser degree of narrowness for allophonic transcription (that is, indicating only some features of allophones, not all of them), the very narrow allophonic transcription will look more like the phonetic (broad) transcription (#1) and less like the impressionistic narrow transcription (#2).

All of this is explained and written on the same page 29 of the book you're quoting in the question: Handbook of the International Phonetic Association: A Guide to the Use of the International Phonetic Alphabet, International Phonetic Association, Cambridge University Press, 1999.