(i) [as conclusion of "what is mathematics/mathematicians?" discussion] A mathematician is someone who sees opportunities for doing mathematics.
(ii) Being a research mathematician is akin to being a writer or an artist; any glamour that's apparent to outsiders fades quickly in the face of [the reality]. Your satisfaction must come from the high you get when you suddenly, for the first time, understand the problem you're working on .....
(iii) As your career develops, the worldwide mathematical community will be increasingly important to you. You will become part of it, and then you will have a home in every city on Earth.
A nice introductory chapter extols the breadth of uses of mathematics, and could be used by Math Dept web sites to recruit majors. There is also a well-balanced "pure or applied" chapter. But the rest of the book identifies "mathematics" as "theorem-proof mathematics", as illustrated by quotes such as
(i) There has been a spate of popular math books in recent years ...... there are even books on the applications of mathematics.
(ii) The true mathematician is not satisfied until the statement is proved.
This attitude is irritating to those of us who regard this identification as akin to identifying "visual art" with oil painting. What matters isn't the tools, it's how competently and creatively you use them.
A final irreverent thought: instead of A Mathematician's Apology, wouldn't The Screwtape Letters have made a more intriguing model?