Learn English – Pronunciation guide for foreign words: Single or double quotation marks

Here is the longer version of the MLA style advice cited in your first quotation:

k. Quotation marks. Enclose in double quotation marks words to which attention is being directed (e.g., words purposely misused or used in a special sense, words referred to as words, and parenthetical English translations of words or phrases from another language). Note, however, that words used as examples in linguistic studies are [italicized] and not enclosed in double quotation marks (see §10h). Use single quotation marks for definitions or translations that appear without intervening punctuation (e.g., ainsi ‘thus’). ...

Crucially, and unlike in your rendition of the final sentence of this advice, the word ainsi is italicized in the MLA guidance. I take this advice as endorsing the following punctuation (if your original wording had been as given below):

The Japanese word Nagasaki ‘na+ga+sa+ki’ means "long cape."

The justification for the single quotation marks at na+ga+sa+ki is that it appears contiguously to the word Nagasaki, as if it were a consecutive translation. That is a special case for which MLA has carved out the italics 'slanted typeface' exception to the normal rule that words used as words should either be set in double quotation marks (in most academic settings) or in italics (in linguistics studies).

The Chicago Manual of Style, sixteenth edition (2010) does not fully agree with MLA on the specific issue of punctuating a foreign word that is immediately followed by a translation, although it does accept the MLA's view of how to handle such juxtapositions in the specific fields of linguistic and phonetic studies:

7.50 Parentheses and quotation marks for foreign words and phrases. A translation following a foreign word, phrase, or title is enclosed in parentheses or quotation marks. ...

The word she wanted was pécher (to sin), not pêcher (to fish).

The Prakrit word majjao, "the tomcat," may be a dialect version of either of two Sanskrit words: madjaro, "my lover," or marjaro, "the cat" (from the verb mrij, "to wash," because the cat constantly washes itself).

Leonardo Fioravanti's Compendio de i secreti rationali (Compendium of rational secrets) became a best seller.

In linguistic and phonetic studies a definition is often enclosed in single quotation makes with no intervening punctuation; any following punctuation is placed after the closing quotation mark. ...

The gap is narrow between mead 'a beverage' and mead 'a meadow'.

Applied to my revision of your original example, Chicago's advice would support any of the following three forms of punctuation:

The Japanese word Nagasaki (na+ga+sa+ki) means "long cape."

or

The Japanese word Nagasaki, "na+ga+sa+ki," means "long cape."

or

The Japanese word Nagasaki ‘na+ga+sa+ki’ means "long cape."

Again, however, a prerequisite for this advice is the unmediated juxtaposition of the term of interest (in italics) with its definition or, arguably, phonetic expression (in parentheses, double quotation marks, or single quotation marks).

The example you ask about does not satisfy this prerequisite, so the MLA and Chicago guidance. That being the case, I would conclude that the appropriate MLA style, by default, would be to use double quotation marks for the trailing phonetic equivalent:

The Japanese word "Nagasaki" is pronounced as "na+ga+sa+ki."

in most academic settings, and

The Japanese word Nagasaki is pronounced as "na+ga+sa+ki."

in linguistics studies. Chicago refers italics to quotation marks, so its preferred handling of your original example would probably be this:

The Japanese word Nagasaki is pronounced as "na+ga+sa+ki."