Why do we have 2 representations for load (shown below) in a power amplifier?
As you can see in the litte graph included in your first schematic there is a current offset ("bias level") for the load current. 

I.e.


current is always positive and 
even if there is no input signal there will be some quiescent current. 


If your load is e.g. a speaker you don't want to have any current going through the speaker if there is silence.
That can be accomplished by AC coupling the speaker like shown in the second schematic.

That way 


current is centered at 0A (it may become positive and negative)
if there is no input signal there will be no current through the  load (no quiescent current through the load)