But that's what the article was all about - implying that 'informal' is not tantamount to 'grammatically incorrect' (by providing 'there's +plural' as an example). Btw, I think there's a bit of discrepancy in conceptions here, so to clarify this: what do you consider a 'spoken' and, for that matter, a 'written' language? For me the latter is nothing more than the representation of the language by means of script/writing system; and the former is what we utter through the mouth. It's not that they are different - grammatically or lexically. What you speak, you can always write as well and vice versa... right?
Well, I do agree that we should speak about formal and informal language, since written can also be informal, etc. Howecer, I do not agree that they are not different, the register is different and thus the way we express something.
What IS sometimes different though, and what the discussion is about is colloquial language vs. formal language. There are many words/expressions which one is unlikely to ever use when, say, talking at a business meeting, applying for a job or speaking to someone really important. However, it's not like people would raise their eyebrows and say that you are not speaking properly if you would use that language in the circle of your friends or at home. For example: some expresions like "just for the heck of it", "catch some Z's" or "slow on the uptake" or "to put the moves on someone" or "put in one's two cents", "greatest thing since sliced bread" ,"go down the drain". The same could also apply to the slang words: (AmE) hunky-dory, honcho, slammer, chintzy, etc. That is something that I consider to be an "informal language".
So, if I may ask you, what do you think is wrong with those expressions/words?
I'm not saying that something should be wrong with those expressions. It's just the matter of when it is appropriate to use them (different type of register). Besides, those are collocations, which has nothing to do with 'bad grammar' as you call it.
Grammatical incorrectness is something completely different, e.g. there's + a plural noun 
Well, the professor of linguistics says the first one is not. But we can just "agree to disagree" ;-).
I still disagree and I think I will never be able to agree with that kind of combination ;-)
So, as I understood you, here's what you are trying to say:
Following the rules = good grammar, yes formal language, not necessarily. Informal language is (or should be) also primarily grammatically correct, but there's a difference in the way we express ourselves, the so called register; there are more differences in vocabulary pronunciation than in grammar as such; this may also be the reason that we feel more freedom to decline from grammatical rules
Informal language= bad grammar only sometimes
Not following the rules = informal language. no, exceptions or 'bad grammar'
Even if you're right about the second statement, the third one still doesn't work as a logical deduction stemming from the previous two statements above, IMO.
Where did I say that?
My main point was, that just because a certain phrase/word is mainly or exclusively used in the informal language, does not mean that it's bad grammatically.
At least we agree in one point.
Usage of an obviously incorrect grammar, errors. Mistakes that you usually correct when teaching. Non-native speakers tend to speak that way.
What is obviously incorrect grammar? To me there's + a pl. noun is! I often correct this one. And at the same time I tell my students that this combination is used in informal language, it's also where they have probably picked it up; American films...
P.S. Here's also a good sentence to consider: "Everyone has their own lifestyle".
Would you say to your students, that it's ungrammatical and thus, it's not the way they supposed to say it - since, obviously, 'everyone' is singular and it doesn't conform with the possessive pronoun.
