Here, we are comparing two options:
a) 山田さんは英語の先生をしている。/ しています。
b) 山田さんは英語の先生だ。/ です。
Let's focus on (a) first. If you check this link, you'd see that it says:
This roughly translates to "to play a role." In other words, you can translate (a) as "Yamada-san plays the role of a teacher." He may or may not be professional/licensed teacher in the strictest sense of the word, but he acts as a teacher.
Now, let's look at (b). From this link, です has this definition
This translates "to express a conclusion or decision." We see that だ or です is used in presenting that something--a concrete or abstract fact--is. This is actually a conclusive statement (as opposed to expressions such as でしょう・だろう or かもしれません・かもしれない) and therefore shows certainty. In other words, this is the most direct translation of "Yamada-san is an English teacher."
Now, why did the book's authors choose the translation "Yamada-san is an English teacher." for "山田さんは英語の先生をしている." I think this is because "Yamada-san is an English teacher." sounds more direct and natural than "Yamada-san plays the role of a teacher."