Q:

New Yorker Dieresis Rule; prosaic, unionized?

There are lots of informal references to the traditional / "New Yorker" style of using diereses to disambiguate runs of vowels, however I have yet to find a definitive guide.
See, for example:

http://www.newyorker.com/online/blogs/culture/2012/04/the-curse-of-the-diaeresis.html
http://en.wikipedia.org/wiki/Diaeresis_(diacritic)
What is the standard rule for using or not using hyphen and diaeresis on the words like reelect , reexamine, and cooperate?

My major question is: Is there a definitive reference that one can consult for proper traditional use of the dieresis in English? The accepted answer in last link above outlines some cases (e.g., no dieresis in "coalesce" or "react") but notably does not provide a reference to an authoritative source.
My minor questions are:

Is it appropriate to use a dieresis in "prosaïc?"
How does one most correctly differentiate the two words spelled "unionized" ("union-ized" about the formation of a group or "un-ionized" about how many free electrons a material has)? Assume for a moment that you have to write both within the same document.

A:

There are lots of informal references to the traditional / "New Yorker" style of using diereses to disambiguate runs of vowels, however I have yet to find a definitive guide.

The definitive guide to the "New Yorker" style, is that you submit a piece to the New Yorker, and if it is accepted for publication it will be cast into that orthography whether you had originally attempted it or not.
The earlier traditional use of diæreses do not have a definitive guide, because it is traditional, and as such was never consistent.
It also isn't really all that traditional, and never dominated English orthography, though it certainly was once more common than before: It was a traditional way of writing, not the traditional way.
And, to dash your hopes of finding a definitive guide further, it was, and remains, used to different degrees by some than by others.
Today, it would still be very common to use a diæresis with a proper noun, like Zoë, Chloë, or Boötes. Likewise with related proper adjectives.
Overlapping with that, it would be much more common to use it in a foreign loan-word that had the diæresis in the original spelling (likewise its identical cousin the umlaut). This though raises another question; how long is a word in English before it is no longer a loan-word? It also brings up a point about how the language has changed, for not only have loan-words become more and more considered to be fully English, but also writers' sensitivity to the foreignness of loan-words has declined so the average scribe is less likely to choose to italicise in a given case than they would a couple of centuries ago. Of such words, perhaps naïve and its relatives (naïf, naïvety, etc.) may be the only word both widely known as English and widely retaining the diacritic it brought with it from France even though it is now 400 years in English use. Perhaps the very fact that it is learnt of relatively early in a child's developing vocabulary that makes it retain the mark; children learn it as "the one with the dots" just as they learn fo'c's'le as "the one with all the apostrophes" and so a general tendency to remove diacritics is resisted just as the general tendency to reduce apostrophes to a ration of one per word.
At the other extreme, those who did (or remain) fond of its use, have no logical reason for avoiding it in react and reaction, beyond the simple fact that it is so commonly seen without one that even the most given to affected feel slightly silly using it.
In-between, some would use it where there was primary stress on the second vowel, and some more widely.
Similarly, many (relatively speaking) would use it where the word was formed by joining two parts, particularly where another might be tempted to hyphenate.
Many would take how likely they thought mispronunciation would be without it.
There is no doubt some influence too from the lack of such a mark in the Latin source; it coming from Latin already joined rather than combined from re- and act in English, but that also applies to some words where one does see it: It's an influence perhaps, but not a self-sufficient reason.

Is it appropriate to use a dieresis in "prosaïc".

It would be technically defensible on the basis of the French prosaïque. Some would only use the diæresis when the primary stress is on the second syllable, and so not use it. Some who would tend to use the diæresis anyway still wouldn't as it is so well-known without and not (unlike the also well-known coöperation for example) formed from combining two parts.
The New Yorker, incidentally, spells it prosaic.

How does one most correctly differentiate the two words spelled "unionized" ("union-ized" about the formation of a group or "un-ionized" about how many free electrons a material has)?

From context.

Assume for a moment that you have to write both within the same document.

As a rule even in the same document the two adjectives are unlikely to be applied to very similar subjects or objects, though certainly if workers are finding themselves ionised on a regular basis they definitely need to organise into a trade union.
As such, it's unlikely to cause that much difficulty here.
If though, you really wanted to distinguish them orthographically, then the diæresis is not going to be of any help. Note that the hyphen is allowed, albeit unpopular, in one but not the other. Hence:

"Organised into a trade union": unionised/unionized
"Not ionised": un-ionised/un-ionized
"Not made more Ionian": un-Ionised/un-Ionized or even un-Ionicised/un-Ionicized.

