What is the definition of Amplifier?

What exactly does the term 'amplifier' come to our mind?
The general assumption, Amplifier is the device that increases the voltage/current amplitude of the signal without altering the nature of the signal.
From the above sentence, it states the observation of the amplifier. But, what exactly the amplifier does?
Before thinking about the exact definition of an amplifier, below shown are the two different networks.
 

                                                                                                                                                                    

The above two circuits which are different coincidentally perform the same operation i.e; increasing the strength or amplitude of a signal. But what differs them? 

Coming directly to the details, In transformers(step-up), we see that

                                          

                                           V2/V1=N and I1/I2=N

which in-turn P1=P2.

The power remains the same before and after enhancing the voltage levels.

But when it comes to an amplifier, the power value gets enhanced. 

So, Let me ask you again? What is the definition of an amplifier?

" The amplifier is an electronic device which enhances the amplitude of the power."