When the word ‘drawing’ is pronounced as /'drɔːrɪŋ/, is that R called intrusive? Is such pronunciation colloquial and unacceptable for formal address?
For people who use the term "intrusive r", it refers to linking /r/ whenever it occurs in a context where the letter R isn't written. It doesn't make a difference whether it is between words (as in saw it) or within a word (as in drawing).
Asking whether "intrusive r" is "colloquial and unacceptable for formal address" is a separate question. Pronouncing "intrusive r"s was traditionally viewed as incorrect but its frequency in current non-rhotic Southern British English accents is high enough that it doesn't make too much sense to call it "unacceptable" if you're trying to describe actual usage.
Yes, an r sound in the middle of a word can be called an "intrusive R."
Tom McLaughlin, a columnist for the Frederick News Post, said he's been on a near lifelong quest to locate the origin of the pronunciation, and he sent me an essay of his headlined " 'WaRshington' Just Sounds Right" to prove it.
So, what do we have here?
Linguists would call it -- in fact, do call it -- an "intrusive R." That's an R that's stuck somewhere unexpected. It happens, said Michael Montgomery, an emeritus professor of linguistics from the University of South Carolina, when the tongue "anticipates" the "sh" sound in "Washington" and curls a little bit.
But this R in wa(r)shington seems to be a different phenomenon than the R in draw(r)ing. The former is a feature of what's called the midland accent.
The [midland] accent can be found in the swath of the country that extends west from Washington, taking in Maryland; southern Pennsylvania; West Virginia; parts of Virginia; southern Ohio, Indiana and Illinois; most of Missouri; and Kentucky, Tennessee, Arkansas, Oklahoma, much of Kansas and west Texas.
The midland accent is rhotic. Usually, the term "intrusive R" is used when describing non-rhotic accents. In a non-rhotic accent, an "intrusive R" is an R that's inserted between a word that ends in a vowel and one that begins in a vowel. These are also called "linking Rs." But not all linking Rs are intrusive.
Consider the following patterns:
In many non-rhotic accents, a linking /r/ is used after some vowels before other vowels. I'll use an R to show where this happens. The vowels I use are those from RP
A staR is born
faR and wide
The ShahR of Persion
went to a spaR and had a sauna
pouR it out
lawR and order
TheRe and then
FaiR and foul
fuR and feather
wateR and oil
RebeccaR and John
HeRe and there
BeeR and cigarettes
In the first set (the /ɑː/s), a non-rhotic accent would pronounce star /stɑː/, except when it's followed by a vowel, in which case it's /stɑːɹ/. That's a linking R, but it's not intrusive, because it's "supposed" to be there. On the other hand, the R in "went to a spaR and..." is intrusive.
Now look at the second (/ɔ/) set. Draw (/dɹɔː/) transforms into draw(r)ing (/ˈdɹɔɹɪŋ(ɡ)/) following the same pattern as law (/lɔː/) -> law(r) (/lɔɹ/) and order!
So, to answer your question with the specific example, yes, the R in draw(r)ing is an "intrusive R."