Learn English – Is it correct to say “ I have been working out lately”

To some extent, this is a general reference question because there are numerous pages on the Internet that discuss adverb place, e.g., this one and this one. However, the question is still an interesting one because there are frequent disputes about specific placements of specific adverbs and whether different placements change the meaning of specific sentences.

The first problem with what you're taught about where adverbs should be placed is that it depends upon the type of adverb and, sometimes, on the specific sentence and specific context of the sentence. For example, most native speakers of English would argue that the following two sentences mean exactly the same thing:

I only eat ducks.
I eat only ducks.

In these two sentences, where the adverb is placed is more or less a matter of style (personal preference). But if the sentences were modified to read, for example:

I only eat ducks, not shoot them.
I eat only ducks, not shoot them.

the syntax of first one makes more sense to me. I think the style is better. But that may be idiosyncratic.

However, assuming two witches chatting online in a contemporary remake of the Hansel and Gretel:

{1} I only eat children, because adults taste bitter. [NB: comma is optional]
{2} I eat only children, because adults taste bitter. [NB: comma is optional]
{3} I eat children only, because adults taste bitter. [NB: comma is not optional]

Only the third one makes instant sense to me. Most native speakers would probably say that the first and third are semantically equal and that the second is ambiguous because it isn't clear whether only is being used as an adverb or an adjective here.

The third sentence could also be:

{4} I eat children, only because adults taste bitter. [NB: comma is not optional]
{5} I eat children only because adults taste bitter.

in which case the meaning of {4} and {5} are different from {1}-{3} and different from each other. Sentence {4} emphasizes why the witch eats children, and sentence {5} is ambiguous: Does only function as an adverb or and adjective here?

The second problem is that when an adverb is placed at the end of the sentence, it usually modifies the entire sentence, just as when it is placed at the beginning of the sentence. In such cases, however, it's usually set off by a comma. But this is far too extensive a topic to discuss in a single answer here.

Of the example sentences you list, I think I can quickly run isn't really acceptable even though it's grammatical and meaningful. It just doesn't sound natural to me, except, perhaps, for a song.

The third problem is that your example contains only adverbs of manner (-ly adverbs). Other adverbs have different rules. It would be better to provide specific sentences, especially longer and more complex sentences. Then the question will become more meaningful for more users.

Where you put an adverb or adverbial phrase depends on what it is you want to modify.

Before a sentence an adverbial is taken to modify the entire sentence; in other positions its scope may be more restricted.

For instance, in your first example, quietly may be placed at the beginning or after teacher if it modifies the main clause, but if it is placed at the end it will be understood to modify the subordinate clause to finish their game. The same constraints operate on your last example, except leasing ... cars loudly and convincingly makes no sense; consequently you will not be misunderstood if you put loudly and convincingly at the end. (But you should not do so, since this will create a momentary confusion.)

In the second example, which contains only a single clause, the adverbial may be placed at the beginning, after she, or at the end.

The third example is tricky. This sentence must be parsed You [can [speak {freely}], not You [[can speak] {freely}]: freely modifies only the lexical verb speak, not the permission or concession expressed by can, so freely must be placed after speak.