In a 2 tier (client/server architecture) application the designer has a choice of where to put the core functionality of the application. If you put it on the client side and use the server as a data store then it is considered a 'Fat' client. 'Smart' is a new name for a Fat client also sometimes referred to as a 'Rich' client.
Choose the server side for the application logic and the client is considered 'Thin'.
Win Forms and browsers are just user interfaces. The browser concept was originally intended as a universal platform independent thin client pallet, one to solve all the world’s problems.
To press the issue about user interface, it is perfectly proper to have a client server application with a fat client and have no user interface at all, no forms or browser at all. This is common.
The issue seems fuzzy because most often in the fat client model the interface and the application are interwoven together or 'tightly coupled'. There is nothing wrong with that in 2 tier architectures, it only becomes problematic when using an n-tier model.
Browsers (read: thin) are a 'loosely coupled' model, intended only to collect and display the results of processing, not do the processing. In theory, only the user’s inputs are processed by the thin client. Nothing about a Win Form is intrinsically fat or thin, indeed it may be either, depending on where you place the application logic. Clear as mud?