No, I think you're right that directions in which solutions to differential equations are determined by their boundary values are generic.

But I think it is reasonable (and maybe common) to identify causality with well-posedness. So I think that rwallace's definition is salvageable.

I'm confused about GR, though, since I remember one guy who put in caveats to make it clear he was talking about uniqueness, not constructive extension, yet still insisted on spacelike boundary.