I am going to take your question literally: " ...what do YOU call it when..." (emphasis supplied). Where I take your 'you' to be me.
I call it charging (and discharging). When I was in college my teachers and fellow students called it charging (and discharging). When I was at work designing electronics, we called it charging (and discharging). The guys who I rubbed elbows with, who wound their own toroids and built power supplies, called it charging (and discharging). I also (infrequently) heard the term "energize/energizing" used; and (rarely) de-energize. That may not be politically or technically correct; but that's how the guys (and gals) I worked with, who actually made the stuff that actually flew on airplanes and spacecraft (and, indeed, enabled them to fly), talked. Nothing wrong with energize/de-energize; but charging/discharging an inductor is perfectly acceptable vernacular.
Think about it from a systems or macro point of view: With a cap you push current into the device to store energy in an electric field. With an inductor you push current into the device to store energy in a magnetic field. With a battery you push current into the device to store energy in the form of a chemical reaction. Discharging extracts energy from whichever field or form is fundamental to the device. The inductor has the neat attribute that you can extract that energy without reversing current flow; but that fact does not demand an alternative word set to "charging/discharging,"