That statement in the dictionary is mostly accurate for the example sentence it describes, but it is a very poor universal rule.
？ She plays very well the piano.
Generally speaking, this is an order that would not normally be used with this sentence. However, even though it's not idiomatic, it's not actually ungrammatical. (Unless you consider its extreme uncommonness to be a sign of ungrammaticality.)
The essential statement is that we don't put an adverb after a verb, but this is not a good universal rule.
Although adverb+verb is more common than verb+adverb, both are possible.
Google Books Ngram Viewer shows that she quickly ran is about half again as common as she ran quickly:
Interestingly, note that from 1827 to 1994, she ran quickly was more common.
Adding an object to the verb doesn't make an essential difference:
In short, the Cambridge Dictionary guidance in this case is not so much about valid syntax, but about common style and usage.
However, while the statement is mostly accurate with respect the particular example sentence it picked, it's not nearly as accurate with respect to every example of verb+adverb+object usage.
Writers can change the normal order of sentences for stylistic effect, in order to put more emphasis on certain things.
In the example passage from Behind the Times, the normal sentence order was changed in order to put more emphasis on the wound than on the finger.