By the way, what is the difference with reason "register" and "profile" ?

It determines the display behavior of a field. "register" means the field will display as if it would on registration. "profile" means the field will display as if it would on profile view/edit.

Does it mean, it will show only fields with active backend checkboxes, i.e. show on register, show on profile etc.

Yes, it helps determine what parameters will factor into its display.