[Gabe]

Would former PH Staff and former TTT Staff (and murder and deathrun and even RP) all fall under the same resigned rank? 
Yes (ex. All resigned staff would have a simple 'Resigned' tag), unless the admins want to make it more complex and make separate resigned ranks for the server you staffed on (ex. Laced Xanax would have a 'Resigned TTT Staff' and HalfDemonLink would have 'Resigned PH Staff'). 

I think the "Resigned Staff Member" title is enough, no need to complicate things by adding specific servers. Also making a separate rank on the forums would probably be hard given that the forums has the security of a military base.

What about demoted or inactive staff? 
If your rank was stripped because you were demoted, for any reason, you did not resign and should not get the rank. 

I agree, if you get demoted you should not get this rank. Maybe if there was a situation where someone was on the verge of demotion and resigned right before then that can be evaluated case by case.

This rank will be shown on our forums / discord, what about the servers?
I would think that the rank would stay within our forums and discord. But if the upper staff want to connect this onto our server, the resigned rank could have no powers included or have donor powers included. I don't know if that suggestion was a bad one. (Now that I'm thinking of it, I don't think that should happen.)

I don't think it needs to go farther than the forums and maybe discord can be added. It doesn't seem necessary or any reason to add it to the server.  I don't think giving powers would be a good idea, like you said, probably shouldn't happen.

Other than recognition, what else can come with this?
Possibly their own discord channel that can be made so ONLY resigned staff can get into. No staff members can get in, no donors-only can get in, only those with the resigned rank.

Being honest here, I think this is useless and will never be used. Especially if only resigned staff can get into it.