Right, I understand the history (although I'm not sure I'd say that exception don't compose well) and I understand that "algebraic effects" are an attempt at something better. But I don't understand whether they're something that can be precisely defined or just informal terminology for "a better sort thing for dealing with effects".
You can precisely define any particular model, but not all work in the area shares the same model. I think you know about the capability-passing model, which is quite different to the algebraic effects (e.g. row types) models.
The general ideas are:
* effects are handled by handlers (called capabilities in the capability-passing model)
* function signatures describe the effects that are used
* effectful code is written in direct style, not monadic style