Learn English – you cook well; you cook good

I would tell your students it's a "trap word," that is, something they might hear when conversing with native speakers but something that others might find jarring or unacceptable. (English has a handful of these – another that I can think of is "The data is..." vs. "The data are...")

As for Mr. Check New Dictionaries, I cringe when someone is dishing out grammar advice while using "u" for you and "n" for and. That undermines his credibility. Moreover, as for his assertion that it's "valid," he's partly right, and partly wrong.

When I consulted the WordNet 3.0 dictionary, for example, it said:

good ADVERB

(often used as a combining form) in a good or proper or satisfactory manner or to a high standard ('good' is a nonstandard dialectal variant for 'well');
- Example: "the children behaved well"
- Example: "a task well done"
- Example: "the party went well"
- Example: "he slept well"
- Example: "a well-argued thesis"
- Example: "a well-seasoned dish"
- Example: "a well-planned party"
- Example: "the baby can walk pretty good"

So the crux of the matter would be this: is ‘valid’ an accurate synonym for ‘nonstandard dialectal variant’? I don't think so. Like that fellow said, it's a matter of choice and habit – but a lot of habits are bad habits.

The majority of the words that you have written in bold are a type of adverbial (a type of adverb) which is called an disjunct. (whether they are all disjuncts could be argued.).

.Disjuncts allow the speaker/writer to influence the hearer or reader.. They represent the speaker's attitude about what he is going to say.

Here are some examples where the adjunct is written in bold:

Naturally, you are going to go you. = the speaker is certain you will go.

Obviously,I agree with the president. = Its obvious that the speaker will agree.

Of course, she is late. = I think it is to be expected.

Regrettably, he didn't attend the meeting.= I think its sad or too bad that he didn't come.

Surely, you are going to go to school today. = a strong sense of persuasion.

Frankly, I couldn't care less about comma rules. = the speaker really doesn't care.

In each case we use a comma to separate the emotion or attitude of the speaker from the rest of the sentence. Some authorities believe that a comma should be used if the disjunct doesn't flow with the rest of the sentence and in most cases disjuncts are separated by a comma.

There are no hard and fast rules about comma use. and convention plays a role in determining some rules. You learn from experience and even English teachers can get it wrong.

Source Grammar 33 manual U of Saskatchewan, Longman's Dictionary, Guide to Grammar Writing by Charles Darling.