Is current pushed through the circuit or taken by the circuit?
Words like "provides", "accepts", "consumes" are all just euphemisms for the circuit is on and working. It makes speaking about the circuit nicer in English. 

In reality, if you have a voltage source and a load is connected to it, current will flow in the whole circuit. The magnitude of the current is dependent on the magnitude of the voltage and the magnitude of the resistance. 

As you said to get more current flowing either the voltage must be increased while maintaining the same load resistance or the load resistance must be decreased while maintaining the same voltage.