correcting a mistake in Spivak [duplicate]

You are correct. This is one of the few errors in Spivak's Calculus on Manifolds. For this particular exercise, see the following questions:

• Question about Angle-Preserving Operators
• Action of angle-preserving linear transformation on basis vectors

copper.hat's answer to the first question cited in the above used the same counterexample as yours.

For your second question, I don't think there is any conflict. Spivak simply meant that if $T$ is a norm-preserving map, then it automatically enjoys the property of being angle-preserving. He did not say that the converse is true. In other words, norm preservation is a stronger condition than angle preservation. In fact, $T$ is norm-preserving map if and only if its matrix w.r.t. the standard basis is a real orthogonal matrix $Q$, and $T$ is angle-preserving if and only if its matrix w.r.t. the standard basis is $\lambda Q$ for some scalar $\lambda>0$ and some real orthogonal matrix $Q$ (for a proof, see the aforementioned answer by copper.hat).

A possible "true 'version' of this statement the author had in mind" is

Suppose $T(x_i)=\lambda_ix_i$ for some basis $x_1,\dots,x_n$ of $\mathbb R^n$ and numbers $\lambda_1,\dots,\lambda_n.$

• If $T$ is angle preserving, then every $|\lambda_i|=|\lambda_j|.$
• If every $\lambda_i=\lambda_j$, then $T$ is angle preserving.

The second part is trivial. A proof of (the contrapositive of) the first part can be found at https://math.stackexchange.com/a/177042.