Question: What do I do when people on the project team all want custom versions of the same doc for different purposes?
Never, ever have multiple concurrent versions of a document active/released at the same time. Try to find out why they want it first.
If you’re distributing printed docs, see if you can give them a watermarked “DRAFT” of some sort, especially if it’s going outside the company. Make it plain which draft/version you branched from and keep it separate from the main source file. PDFs can also have watermarks or other tags in them, passwords and such, to keep bad people out.
Up-revving a document letter might be misleading because usually “Rev B” is considered the successor to “Rev A” but that’s not true in this case. I guess you could just make unique numbers for every variant of the same doc. No one will ever find that confusing later on.
If you’re in some online system with Wiki pages, maybe there’s some way to create multiple links but share them selectively so each requestor thinks that THEIR branch is the “right” one. If your project team isn’t talking amongst themselves already, chances are you can get away with this one for a long time.
Think “what if some other writer has to pick this up in a month, will they be able to identify the correct source file?” Two ways to track this would be the Notes section on a Smartsheet (add a line for the bastard child) and possibly, a sticky note on a printout if you store hard copies in a file drawer.
Also think, “how would I explain this to the project manager or another team member without sounding like an idiot?” and if your face turns red trying to explain the reasoning, it’s probably a bad reason and should be escalated until common sense returns.