What is the name for "pronunciation spelling"?

Solution 1:

There is a universal (more or less) system for recording human speech sounds.
It's called the International Phonetic Alphabet, or IPA.   (Note: That's Phonetic, not Phonemic.)
This contains all the sounds that occur in any human language.
Naturally, no human language has all of them.
No human language even has most of them.

Individual languages use only some of those sounds, and representations of words in individual languages use a special selection of these symbols -- different for every language -- called "phonemes". Usually the symbols used are taken from the IPA, but they represent only the sounds and their distribution and pronunciation in that language.

For instance, this is the American English phonemic system, from Kenyon and Knott.

English dictionaries published in the United States normally don't use phonemic pronunciation, preferring the system invented by Noah Webster, based on spelling instead of phonetics. Webster was a spelling reformer, and believed that the traditional English spelling could be used to indicate phonetics. He was wrong, but that hasn't influenced American dictionaries.

English dictionaries published elsewhere, or bilingual dictionaries, or dictionaries intended for language learners, normally do use standard phonemic transcription. If you have a bilingual dictionary, look at the pronunciations in the English part; they will normally use either the system of Kenyon and Knott (American) or a system of RP (UK), which has some differences from American. This is because English speakers normally pay no attention to the pronunciations in the English part, but English learners do, and they need accuracy.

The example cited

\pȯrt-ˈman-(ˌ)tō\

is the style used by Merriam-Webster, based on the original nonphonemic transcription.
In Kenyon and Knott's system, it would be

/portmænto/

and in RP it would be

/poətmæntəʊ/

Regardless of what the Wikipedia entry above says, the M-W system is not phonemic.

Solution 2:

Pronunciation Respelling -- "non-phonemic" or "newspaper" systems

Pronunciation respelling is a notation used to convey the pronunciation of words, in a language, such as English, which does not have a phonemic orthography. … "non-phonemic" or "newspaper" systems, commonly used in newspapers and other non-technical writings, avoid diacritics and literally "respell" words making use of well-known English words and spelling conventions, even though the resulting system may not have a one-to-one mapping between symbols and sounds.

As an example, the last name of actor Jake Gyllenhaal, written ˈdʒɪlənhɔːl in the IPA, might be written jĭl′·ən·hôl or JIL-ən-hawl in a phonemic system, and Jill-in-hall in a non-phonemic system.