ArduCam ESP8266 and (Not-Mini) OV2640 Camera Module interface?
No, you can't "mix and match" like that. The 8-pin header has I2C and SPI on it for a "cooked" video interface. The 20-pin interface has the "raw" camera CSI interface.

The difference is, the mini version has its own small video processor and FIFO chip on it which processes and buffers the video image allowing you to take a picture then slowly stream that data back through to the slow Arduino at a speed it is comfortable with. The non-mini version doesn't have that and relies on the host controller board having enough power to process a live CSI video signal, which the Arduino cannot do.