Correct definition of terminal alkynes?
You are correct about the absolute numbering of $\ce{CH2=CH-C#CH}$. By the IUPAC priority rules, the alkene has the lower absolute numbering.
However, numbering can also be relative. In your case, the relative numbering 1,2 refer to either of the following:

relative position from the end of the chain, where the chain end is #1
relative position from the functional group you care about (where that group is #1)
suggesting that a reaction occurs on neighboring positions in a molecule.

That being said, the definition given by Brown et al. is imprecise. A better definition is one of the following:

A terminal alkyne is an alkyne in which one (or both) of the two alkyne carbons is at the end of a chain.
A terminal alkyne is an alkyne $\ce{RC#CR}$ in which one or both $\ce{R}=\ce{H}$