Well, with Progress Quest you name your character and then it just adventured without you. You had absolutely no control over anything, except to pause or unpause. I'd call that a non-game "toy". A lot of the dress-up Flash games are toys.
A Tomagotchi was a handheld virtual pet device. You had to feed it and play with it and clean up after it. I'd call that a "virtual pet" which is more than a "toy" but it's borderline on "game". Everything about it is toylike, but you do have control over it, so I can't just outright call it a toy. So "virtual pet". You see some Flash games like this too. There was one where you could do different stuff to kill your pet and that was the whole point to the thing.
Travian is a "game", because you have interesting choices to make, the outcome is uncertain, and you have various controls over it. It is a "massively multiplayer" game because it involves thousands of other players sharing a play environment, and their choices impact each other. It does continue working while you're gone, to complete your orders, and the pay account can set up build queues so you can leave it alone for longer and still have to carry out your orders.
Starcraft is a "game" as above. It's a "single player game" because you can play alone vs the AI. It's also a "multiplayer game" because you can play against / with a few other players at once. You can also set up a map where all the sides are AI, and they fight it out themselves. In this configuration it's a "toy" - it is no longer a game in that configuration because you do not make interesting choices or have control over it.
So if Godville has a mode where you can interact with your character, and you can make interesting choices, and the outcome is uncertain, then you have a "single-player game". The online rankings don't change that. If there is a mode where the character acts without your input, then that is a "toy".
EDIT: I just want to make it clear though, that both modes of Godville are covered by "single-player game" or "toy". There is no need for a "zero-player game" classification, and in fact such a classification is impossible because a game designed to include no players is a "toy". The closest you'd come is a "zero-player toy" which is redundant.
EDIT2: The whole quirky name thing just has me irritated. Makes me think of a sleazy young business-type guy trying to sell $1 sunglasses for $50 at a convention booth or street corner.
EDIT3: Look at me! I can call Mario Kart a Calculating Unit Number Taker! Or a Physical Entity Non-Intelligence Simulator! How in the world does that help anything? Who cares, it might move a few more units!