Can I record my performances with DJM-REC while using the 1/4" Jack in the MULTI I/O section?

Yes. You can use DJM-REC regardless of the [SOURCE] setting in the MULTI I/O section.
You can record your performances using DJM-REC even if you use an external effects unit connected to the 1/4" Jack in the MULTI I/O section.
Note: iPhones and iPads with a Lightning port won't work; you'll need to use an iPhone or iPad with a USB Type-C port instead.