For pay then, I’d surmise that it would be “hiring”. Now of course, they have to “select” you from a gaggle of candidates, first. But if they’re going to pay for your services, then that would meet the definition of hiring.
Being hired therefore gets you on the payroll and all that goes along with it.
If the internship were paid only in experience, then there would, in effect, not be a hiring process, merely the selection.