fix(logging): add API response timestamp and fix request timestamp timing

Previously:
- REQUEST INFO timestamp was captured at log write time (not request arrival)
- API RESPONSE had NO timestamp at all

This fix:
- Captures REQUEST INFO timestamp when request first arrives
- Adds API RESPONSE timestamp when upstream response arrives

Changes:
- Add Timestamp field to RequestInfo, set at middleware initialization
- Set API_RESPONSE_TIMESTAMP in appendAPIResponse() and gemini handler
- Pass timestamps through logging chain to writeNonStreamingLog()
- Add timestamp output to API RESPONSE section

This enables accurate measurement of backend response latency in error logs.

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

@@ -125,6 +125,7 @@ func (h *GeminiCLIAPIHandler) CLIHandler(c *gin.Context) {
 			return
 		}
 		_, _ = c.Writer.Write(output)
+		c.Set("API_RESPONSE_TIMESTAMP", time.Now())
 		c.Set("API_RESPONSE", output)
 	}
 }

sdk/api/handlers/handlers.go

@@ -361,6 +361,11 @@ func appendAPIResponse(c *gin.Context, data []byte) {
 		return
 	}
 
+	// Capture timestamp on first API response
+	if _, exists := c.Get("API_RESPONSE_TIMESTAMP"); !exists {
+		c.Set("API_RESPONSE_TIMESTAMP", time.Now())
+	}
+
 	if existing, exists := c.Get("API_RESPONSE"); exists {
 		if existingBytes, ok := existing.([]byte); ok && len(existingBytes) > 0 {
 			combined := make([]byte, 0, len(existingBytes)+len(data)+1)