“A mutual friend”, "A common friend" or “A friend in common”?

When I was a student of EFL, I was taught the word "mutual" implies a reciprocal relationship where X does to Y what Y does to X. If John admires Peter as much as Peter admires John, we can say their admiration is mutual. That said, can I say that Charles and I have a mutual friend ?

According to the following source the expression mutual friend, despite language critics objections, is commonly used and accepted.

Mutual: Usage Note,

Mutual is used to describe a reciprocal relationship between two or more people or things. Thus their mutual animosity means "their animosity for each other" or "the animosity between them," and a mutual defense treaty is one in which each party agrees to come to the defense of the other. But many people also use mutual to mean "shared in common," as in The bill serves the mutual interests of management and labor. This usage is perhaps most familiar in the expression our mutual friend, which was widespread even before Charles Dickens used it as the title of a novel. While some language critics have objected to this usage because it does not include the notion of reciprocity, it appears in the writing of some of our greatest authors, including Shakespeare, Edmund Burke, George Eliot, and James Joyce, and it continues to be used by well-respected writers today.

Also Ngram shows a more common use of the expression usual friends compared to the other two suggested alternatives.

Source: The American Heritage® Dictionary of the English Language.

Yes, if you and Charles are both friends with Alice, you can refer to Alice as a mutual friend. This allows an implication that you are a friend of Alice, and Peter is a friend of Alice, but you are not necessarily a friend of Peter.

The designation "a common friend" means nearly the same, but has a small risk that someone might wonder if there is a better friend who is better than just common, as in "Eddie is a common friend, but John is in a class by himself."