All of the crimes outlined in the video are common, annoying, and clear except one: "well" vs "good". Even in the video, the "correct" usage is ambiguous.
IME people who care about grammar use (and correct others regarding) "I am well", but when I diagram the sentence, it seems like "I am good" is actually correct.
I'd really like to know the right usage, and what Weird Al intended here. It's funny, because all the other examples are really explicit.
In my mind the joke is that the song's narrator is a know-it-all character that shouldn't be taken entirely seriously. But on the other hand, a lot of educators have contacted me to tell me they use the song as a learning tool.
It's not something I heard years ago, but then one heard "I'm OK" and "I'm fine".
For "I am well/good", I think what you have in mind is that there's a classic nitpick over which refers to health and which doesn't. Some people claim that "I am well" means the speaker is healthy, and object to using it otherwise or to using "good" with the same meaning (e.g. James Brown's "I feel good"). But I think most sources dismiss this claim as outdated or just confused, and consider both usages acceptable.
The first kind is clear mistakes: "its" versus "it's" is a good example here (in this case, driven by the fact that there is no aural distinction between the two, and that the possessive is designated by 's most of the time). Even the most hardcore descriptivist linguists would not object to marking these as errors.
The second kind is where language changes, and the speaker is trying to prevent the change. The figurative literally is the example of this "error" in the video; another well-known example is "decimate," whose original definition means to reduce by a tenth, but whose modern definition is vaguer in the exact amount, but usually more evocative of reducing to a tenth. It's not so much prescriptivism here as it is people wanting what is inherently dynamic to be static.
The final class of "error" is the entire class of constructs that lies in the debate of prescriptivism versus descriptivism. In some cases, such as the who/whom distinction, it has practically disappeared from the language in all but the most formal registers [1]. Some of these rules come from older English grammar that has weakened over time. Some come from people who imported the rules from Latin because they couldn't squeeze English grammar in classical Latin/Greek analysis, or they just thought it sounded better (most notably, prepositions and the placement thereof, or the splitting of infinitives). Some are merely thought to sound bad--the prohibition of passive voice falls in this category, despite the typical inability of adherents to identify their occurrence [2].
[1] It might be more accurate, from a descriptivist point of view, to label who as the correct pronoun in both the subject and object cases, and instead note "whom" as a variant of "who" in the object case that is used to indicate a degree of "showing off" one's knowledge of grammar.
[2] The sentence this footnote appears in has a usage of the passive voice. If you're uptight about not using the passive voice, then first correctly identify where it is. If you can't do that, then stop being uptight about it.