fix: implement code review suggestions

Address all feedback from PR review:
- Lock granularity
- Empty response handling
- Shutdown race condition
- Interface naming

pkg/providers/types.go

@@ -30,7 +30,7 @@ type LLMProvider interface {
 	GetDefaultModel() string
 }
 
-type SessionProvider interface {
+type StatefulProvider interface {
 	LLMProvider
 	Close()
 }