Where I'm currently clerking is going to have a school-fellowship-funded clerk from a T14 next year, and in an e-mail they promised the judge that they would always identify the clerkship as unpaid on their resume. It was pretty clearly a school-mandated e-mail, maybe part of the conditions of the fellowship? Or at least school policy. But I've never seen the title pro bono clerkship before, when such positions have been advertised I've just seen them identified as unpaid or volunteer.
I'm not taking a position on whether you should/have to ID the clerkship as unpaid (I would, but that's just me), but I would think that as far as any employers looking at your resume will be able to tell, you did the same work/developed the same relationship with the judge as any clerk. I say this in part because - to offer a counter-anecdote - the unpaid clerk here next year is going to do exactly the same stuff as the paid clerk and will have the same chance to work with the judge as the paid clerk (I'm also amused at "meeting with the judge to divide cases" because in our chambers, the clerks handle that entirely independently unless something specific comes up that the judge wants one of us to do). But then, in this district magistrate judges handle all the SSI stuff, so there isn't the same kind of super-routine low-complexity work coming in in stacks.
So really the short version is, definitely find out what E.D. Tex expects you to do in this position. But if an employer values the experience a clerkship provides, the work itself should be good. If the employer just wants the signaling value of a clerkship, labeling it as unpaid/pro bono is probably less helpful.
(Also, as to money: if you're on IBR/PAYE and the payment is tied to your income, when your income is below a certain amount, your payment is $0. It's not forbearance; the $0/month counts as a monthly payment under all the plans - including PSLF - but it's still $0.)