Your reasoning is correct if we assume that
> **the person whose name begins with the same letter as that of the name of the person who will tee off first**
is to be understood as different from
> **the person who will tee off first**.
This isn’t the only possible interpretation: if, for example, Diana tees off first, her name does begin with the same letter as that of the name of the person who tees off first!
However, without this assumption the problem is insoluble, so presumably we’re intended to make it.