What actually is the output current of a boost conveter?
As your current graphs show, the inductor current is not constant, so the concept of 'the inductor current' being a single (time-independent) value is not valid.

What is valid though, is that the inductor current has no sudden changes (the first derivative is always finite). This is a fundamental property of an inductor, and it is illustrated in your 3d graph: the inductor current is a sawtooth.

When you talk about 'output current' you must be careful what you mean: if you say inductor_current = output_current you mean the current going into the (load + output_capacitor). 

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(after question update)

CCM means that there is always a current through the inductor. That means that the current fed into the load+capacitor ramps down to a certain value, and then abruptly drops to 0. 

DCM means that at some point(s) in time there is no current through the inductor. That means that the current fed into the load+capacitor ramps down to 0. 

Hence DCM/CCM can be observed both at the inductor and at the load+capacitor (and at the input too).