Learn English – Which version of this sentence with a linking adverb is grammatically acceptable

Of your "really" examples:

1. I really don't care

2. I don't really care

3. I don't care really

I would claim that (3) needs a comma and it has the same meaning as (2).

I don't care, really

(2) and (3), therefore, roughly mean, "I am mostly apathetic about this." (1) means, "I very much do not care about this. [Stop bothering me.]"

There isn't anything inherently ungrammatical about your "quite" examples but they aren't typically used in American English. The most common would be (2): "I don't quite care [but I almost do.]" I've never encountered the other two.

If you look at "quite" in other contexts, this pattern plays out:

1. I quite didn't make the meeting on time.

2. I didn't quite make the meeting on time.

3. I didn't make the meeting quite on time.

4. I didn't make the meeting on time quite.

The only examples that are used in American English are (2) and (3). Contrast this with "really":

1. I really didn't make the meeting on time.

2. I didn't really make the meeting on time.

3. I didn't make the meeting really on time.

4. I didn't make the meeting on time, really.

(1) and (2) are common, (3) would sound weird but I can imagine someone saying it and (4) only works with the commma.

In his text The Syntactic Phenomena of English, McCawley gives a taxonomy of adverbs based on what they modify. While most adverbs do not modify verbs, according to him, he argues that degree adverbs, like "completely", do modify verbs. I would suppose that this also applies to "substantially". Furthermore, McCawley says that in general, modifiers tend to occur either immediately before or immediately after what they modify.

These general grounds would lead us to expect that "substantially" ought to occur immediately before or immediately after the main verb of a sentence. However, degree adverbs can ordinarily also occur at the end of the verb phrase: "He missed the target substantially." And there is another rule of English that a clause adverb does not occur between a transitive verb and its direct object: *"He missed substantially the target."