refactor: standardize constant naming and improve file-based auth handling

- Renamed constants from uppercase to CamelCase for consistency.
- Replaced redundant file-based auth handling logic with the new `util.CountAuthFiles` helper.
- Fixed various error-handling inconsistencies and enhanced robustness in file operations.
- Streamlined auth client reload logic in server and watcher components.
- Applied minor code readability improvements across multiple packages.

internal/api/handlers/claude/code_handlers.go

@@ -43,7 +43,7 @@ func NewClaudeCodeAPIHandler(apiHandlers *handlers.BaseAPIHandler) *ClaudeCodeAP
 
 // HandlerType returns the identifier for this handler implementation.
 func (h *ClaudeCodeAPIHandler) HandlerType() string {
-	return CLAUDE
+	return Claude
 }
 
 // Models returns a list of models supported by this handler.

internal/api/handlers/gemini/gemini-cli_handlers.go

@@ -38,7 +38,7 @@ func NewGeminiCLIAPIHandler(apiHandlers *handlers.BaseAPIHandler) *GeminiCLIAPIH
 
 // HandlerType returns the type of this handler.
 func (h *GeminiCLIAPIHandler) HandlerType() string {
-	return GEMINICLI
+	return GeminiCLI
 }
 
 // Models returns a list of models supported by this handler.

internal/api/handlers/gemini/gemini_handlers.go

@@ -38,7 +38,7 @@ func NewGeminiAPIHandler(apiHandlers *handlers.BaseAPIHandler) *GeminiAPIHandler
 
 // HandlerType returns the identifier for this handler implementation.
 func (h *GeminiAPIHandler) HandlerType() string {
-	return GEMINI
+	return Gemini
 }
 
 // Models returns the Gemini-compatible model metadata supported by this handler.

internal/api/handlers/management/auth_files.go

@@ -147,7 +147,7 @@ func (h *Handler) UploadAuthFile(c *gin.Context) {
 		c.JSON(500, gin.H{"error": fmt.Sprintf("failed to write file: %v", errWrite)})
 		return
 	}
-	if err := h.registerAuthFromFile(ctx, dst, data); err != nil {
+	if err = h.registerAuthFromFile(ctx, dst, data); err != nil {
 		c.JSON(500, gin.H{"error": err.Error()})
 		return
 	}

internal/api/handlers/openai/openai_handlers.go

@@ -44,7 +44,7 @@ func NewOpenAIAPIHandler(apiHandlers *handlers.BaseAPIHandler) *OpenAIAPIHandler
 
 // HandlerType returns the identifier for this handler implementation.
 func (h *OpenAIAPIHandler) HandlerType() string {
-	return OPENAI
+	return OpenAI
 }
 
 // Models returns the OpenAI-compatible model metadata supported by this handler.

internal/api/handlers/openai/openai_responses_handlers.go

@@ -43,7 +43,7 @@ func NewOpenAIResponsesAPIHandler(apiHandlers *handlers.BaseAPIHandler) *OpenAIR
 
 // HandlerType returns the identifier for this handler implementation.
 func (h *OpenAIResponsesAPIHandler) HandlerType() string {
-	return OPENAI_RESPONSE
+	return OpenaiResponse
 }
 
 // Models returns the OpenAIResponses-compatible model metadata supported by this handler.
@@ -161,6 +161,7 @@ func (h *OpenAIResponsesAPIHandler) forwardResponsesStream(c *gin.Context, flush
 			return
 		case chunk, ok := <-data:
 			if !ok {
+				_, _ = c.Writer.Write([]byte("\n"))
 				flusher.Flush()
 				cancel(nil)
 				return