Conceptual Model
Conceptual model is our way to see how the thing might work and how to operate over it. For example, when we see a bike, we can understand how it works in general be modeling it in our mind. Conceptual model is actually defined by 3 parts - affordance, constraints and mappings.Affordance helps to understand how basically approach the thing. Constraints help to see how the best operate over the thing.
Affordance and Contraints helps to make the conceptual model obvious. For example, scissors. There are 2 holes in it. So we can understand that they are to put something inside, like fingers. While constrains help to understand what fingers we need to put inside.
Affordance and constraints helps to build visibility, which is required to understand how the thing works and what it is for. For example, scissors have clear visibility, while handwatches with multiple are not as clear. Obvious conceptual model is required for product, if you want its design to be transparent to customers.
Design model is how the product seen by its designer. User model is how the product seen by its user.