Capacitor current vs supply current?

I'd say they are different, based on the topology in your link.

The supply current can be intermittent, but it is always in the forward direction.  If only one value is given for it, then that probably is the time-averaged value of current from the source.

The capacitor appears to be there to smooth out the current from the inductor.  The current through the inductor ramps up and down (like a triangle wave with a dc offset, see the graph in the link).  If the load is taking constant current, then the capacitor takes up or supplies the slack.  The current through it is an alternating current (triangle wave), and it should be a much lower magnitude than the load or inductor current.

If you happened to "tune" the capacitor where it resonated with the inductor, then bad things could happen, although if there is a closed-loop feedback control system, that might not be an issue.