refactor: improve lifespan context typing and documentation

- Add proper generic parameter for lifespan context type
- Update README with TypedDict example for strong typing
- Fix context variable initialization in server
- Improve property return type safety
- Remove redundant documentation
- Ensure compatibility with existing tests

src/mcp/server/fastmcp/server.py

@@ -100,7 +100,7 @@ class Settings(BaseSettings, Generic[LifespanResultT]):
 
     lifespan: (
         Callable[["FastMCP"], AbstractAsyncContextManager[LifespanResultT]] | None
-    ) = Field(None, description="Lifespan contexte manager")
+    ) = Field(None, description="Lifespan context manager")
 
 
 def lifespan_wrapper(