Model
An extremely overloaded word that it means very different things in different fields. Even within AI research it is used loosely. Throughout this wiki, "model" almost always means a machine learning (ML) model, and usually a language model.
A model is the trained artifact that a machine learning process produces: the patterns it learned, stored as a huge list of numbers called its weights (or "parameters"). The weights are the model and running it means feeding in some input and letting those numbers turn it into an output. (How it comes to learn those patterns is covered under Machine learning.)
A model is not the same as the product built around it, and the difference matters when judging a tool. Think of the model as a car engine and the system as the car built around it. An engine on its own is not something you can drive. It takes a chassis, steering, pedals, brakes, and a dashboard to make it usable. The same engine can go into a well-built, safe car or a shoddy, dangerous one. In the same way, the model is the raw learned predictor at the core (just a set of weights that scores how likely some output is), and the system is everything wrapped around it to make a usable product: the interface you type into, the instructions it is fed behind the scenes, the branding, any added data or filters, and how its output is presented.