I'm gonna simplify and clarify this question, since it has proved popular:
Look at this SO QA ..
Where an ellipsis exists, is there a term for the missing text?
Observe that at first it appears AA was being asked. People answered this at length. Then after some clarification it would appear BB was being asked. Other people answered this at length. Finally it appeared CC was being asked.
The same sort of thing also happens particularly in business – it's thought that the client wants A but after some work on A everyone realises they really meant B .. and so on. (See my earlier over-long edit of this question for an example.)
What's the best term for this particular type of snafu?
There's kind of a taxonomy of fuck-ups in communications, in business, in rhetoric, in discussion: as DB mentions below, "spec creep" is an excellent example of something that would be in such a taxonomy!
What's the best phrase or perhaps single-word here?
Best Answer
That's classic "scope creep". Per Wikipedia:
According to that article, one of the principle causes of scope creep is ambiguous or unclear requirements. In your case, it sounds like your clients are deliberately taking advantage of this (inherent) ambiguity, which I would call disingenuous.
Another way to describe the practice of changing the objectives as you play is "moving the goalposts":
And of course, from the other side, there's the classic "bait and switch".