Proving undecidability of HALT_tm by reduction
Sipser reduces $A_{\mathit{TM}}$ to $\mathit{HALT}_{\mathit{TM}}$. Since $A_{\mathit{TM}}$ is uncomputable, it follows that $\mathit{HALT}_{\mathit{TM}}$ has to be uncomputable. He is using the following general statement:


  If $A$ reduces to $B$ and $B$ is uncomputable, then so is $A$.


You are right that this general statement itself is proved by contradiction. A proof which uses this general statement to prove that a particular problem is undecidable is a proof by reduction – in your case, you prove the undecidability $\mathit{HALT}_{\mathit{TM}}$ by reducing a problem you already know is undecidable to $\mathit{HALT}_{\mathit{TM}}$. In contrast, Sipser's proof that $A_{\mathit{TM}}$ uses a different technique, diagonalization.