Here is the short description of scientific model:
an imperfect or idealized representation of a physical system
And the definition of toy model:
a simplified set of objects and equations relating them so that they can nevertheless be used to understand a mechanism that is also useful in the full, non-simplified theory.
I don't understand the difference between them. Isn't any model used to "understand a mechanism that is also useful in the full, non-simplified theory"? The Ising Model is listed as a toy model, but isn't that natural when modelling a phenomenon: you start from a simple explanation, then improving it by adding more correction? And if it is just a toy model, why bother to expand it to higher dimensions? It even has application in neuroscience. I would say it's quite successful.
IMO even Standard Model can be seen as an toy model. Is that correct? And at what point it is not a toy model anymore? When it can give some quantitative results that agree with experiments? But then any model does have its own limitation, right?