"Proof by Contradiction" Proving a theorem $A \Rightarrow B$ (i.e. if $A$ is true then $B$ is true) by contradiction means:

Assume $B$ is not true, and arrive at $A$ is not true.

Consider a theorem of "if and only if" type: $A \Leftrightarrow B$.
If we prove $A\Rightarrow B$ and $A^c\Rightarrow B^c$ (i.e. if $A$ is not true then $B$ is not true), should we call this proof a proof by contradiction? Is there a precise definition of "Proof by Contradiction"? 
 A: What you have called "proof by contradiction" in you question is more properly called "proof by contrapositive": From $A\implies B$, deduce $B^c\implies A^c$. Your second proof strategy is another instance of this, because, if we know that $A^c\implies B^c$, then, by contrapositive, $B\implies A$.
Proof by contradiction instead refers to the following practice:
To prove $A$, first assume $A^c$, and then deduce something false. Since only false premises imply false conclusions, we can then know that $A$ must be true.
While both techniques involve assuming the opposite of what you want to know, there are important differences between them. Fore example, proof by contrapositive is used specifically when proving implications, while contradiction can be used more generally. Also, various constructivist logic systems treat these two techniques very differently.
A: I think you might have a small misunderstanding between contrapositive and contradiction.
Contradiction is a proof technique to show that a statement $A\implies B$ must be true:

Assume B is not true, and arrive at A is not true.

Contrapositive, on the other hand, is a restatement of your problem, to provide another angle of thought. You show that $A\implies B$, by showing the truth of the negated statement $B^{c} \implies A^{c}$. Then, you can show this statement is true by using either a direct proof or contradiction.
In your question, by asking if you can show the truth of $B\implies A$ by showing $A^{c} \implies B^{c}$, you would not be using contradiction, but rather changing the second component into a contrapositive.
In summary, contradiction is a proof technique, while contrapositive is a different way to think about the same problem.
