Well, the first thing to say is that English majors tend to look at grammar differently from Linguistics majors, so I'm kind of going back and forth between both systems. In linguistics, we look at grammar from a very functional perspective, while classical grammar is more about creating a fixed set of categories and fitting words into those categories as best possible.
Reading
back over that
again, I
probably overemphasized the extent of
this particular rule. We were
strongly, strongly discouraged from using adverbs (notice the irony there) because they tend to water
down your writing
more than they help to emphasize a point.
Instead, we were encouraged to find bolder ways of emphasizing our points (stronger verbs)
instead of
just saying that something is "
really really ridiculously" whatever. We were discouraged from using be verbs (more irony) because that forced us to use active voice. I suspect that she may have been using overemphasis as a teaching tool. If you are forced to write in this particular style, you have to
actually think about the words you're using,
instead of
just writing
how you talk. Either way, I felt like my writing benefited
tremendously. Do I
still write
exactly that way? No, not
exactly. But I feel like I learned the principle
well enough that I've become
really particular about things like diction and syntax.
Now that I've broken
all of those rules, I also feel compelled to say that I
only use these rules in formal settings. So don't judge my writing based on the stuff I've written here

red words - Adverbs that are formed regularly from adjectives by adding an "-ly" suffix. (That said, "really" and "only" are such common words that most English speakers probably don't associate them with "real" and "one.") I'm thinking that these are the ones your English teacher wanted you to avoid (when they modified verbs, at least).
green words - Adverbs not formed regularly from adjectives. (Such adverbs are typically very old in English, although the one formed from a preposition + a noun is more recent.)
blue words - In linguistics we'd call these particles. (Many of them are words that can also be prepositions, but prepositions and particles behave quite differently.) In classical grammar, they're called adverbs because they accompany verbs and because there's no separate category for particles. Linguists consider particles different from adverbs because they behave differently in terms of placement (adverb placement is quite fluid, while particles have to follow verbs, and may follow or precede objects) and because they can significantly alter the meaning of a verb, instead of just modifying it somewhat. I thought I saw another particle when I first read through this, but now I can't find it.
orange words - Words that I'm not sure about. (I'd have to think more about these or poll a bunch of my language-loving friends to see if any of them can think of good tests to run on them.)
purple words - Words that I don't think are adverbs, with explanation:
"instead of" - "Instead," by itself, is pretty obviously an adverb. It's a single word that modifies an entire sentence or clause, which can be interpreted as modifying the main verb in that sentence or clause. You can also grammatically switch it out with something like "however," a word that is unquestionably an adverb.
However (!), when you put "instead" in front of "of," it starts behaving very differently. For starters, prepositional phrases don't modify adverbs, and there's no other nearby verb, noun, or determiner (= linguistics term roughly equivalent to pronouns), for it to modify. And notice that you can't switch it out with the adverb "however" and have the phrase make sense, but you can switch out with the prepositional phrase "in place." (E.g., "Instead of saying X, Y, Z . . ." "In place of saying X, Y, Z . . .") So, I'd argue that "instead" before "of" is a preposition and a noun that happen to be written as one word. (Historically, "instead" was written as two separate words, and the adverb "instead" definitely comes from the phrase "in stead," but in modern English, the adverb "instead" behaves differently from the "instead" in front of "of.") Of course, classical grammar people want each word to fit into one category, and "prepositional phrase" is not an option, so I don't know what they'd do with "instead" in front of "of." But I do know that it fails all the tests I can think of for adverbs.
"all of those rules" - My gut says that "all" is a determiner, which is basically the linguistics category that comprises pronouns (and articles). However, the
OED is trying to convince me that it's an adjective or a noun, which I'm not buying. Regardless, it's failing the tests for adverbs.
"this particular rule" - "This" is a determiner. The
OED calls it (*sigh*) an adjective. For it to be functioning as an adverb, it would have to be in a phrase like "walk this far" or "be this tall."
As for the phrases, you've pulled out what linguists would call inflectional phrases (the phrases starting with an infinitive) and a couple of prepositional phrases. If memory serves, classical grammar is fond of all kinds of special adverbial phrases, and I'm not sure if these would qualify or not. Linguists name a phrase by the function of the head word of the phrase, not by the function of the phrase overall.
Fun times! And I should say that you caught several adverbs I missed in my first reading, so I definitely had the advantage of making you go first. Now I will probably dream of adverbs, tonight.