I think the best way to say that sentence is:
박 과장님한테 전화해서 빨리 사장님께 전화드리라고 말씀드리겠습니다 (or 전하겠습니다).
You're supposed to raise the person you're speaking with more, especially when they are at a higher position as in this case. You can use the usual mid-level polite forms for the other person.
What the book explains is the traditional way of using relative honorifics, like when talking to your grandfather, you don't raise your father since your father is a son to your grandfather (i.e. you tailor the language to grandfather). This used to be the standard way in the old days, such that a daughter refers to her husband with a humbling term like 아비/애비 and a son his wife as 에미 when talking to parents or parents in law. But this practice is rapidly changing in modern times.
First off, very few families live with their parents in the same house these days, so there are fewer and fewer occasions to use such honorifics. And to the new generations, such an old custom may seem too extreme and weird. People nowadays tend to think the universal idea of all humans being equal is a higher principle than showing respect even changing the language just because someone is older or at a higher social position. You certainly cannot do away with honorifics embedded in the language, but the general trend is toward using simpler phrases.
Secondly, we should consider the work environment. Workplaces strive to be efficient, and using such kind of complicated languages goes against this. Most companies want their employees use simple and pleasant language. If 박 과장님 is older and higher than you, using 님 and reasonable honorifics is a sensible thing to do even when talking to the president in my opinion. Using similar language all around is the easiest, and big corporations encourage this (I've read news reports on this). So I personally think your friend is right and your book needs updating. But note that all this may vary with people and places.