You are going to want to use SBUS with that receiver. If you look at the top right corner of the board (in your first picture), it says "SBUS". It will be these three pins that you will connect to your receiver.
Coming from the receiver, there will be 3 wires (typically white/yellow, red and black). The white/yellow line is signal, and that will connect to the SBUS pin. The red is 5v, and the black is ground.
On the X4R, there is a set of 3 pins that are labelled similarly. Those are where you will put the connection to the receiver.
I am not totally familiar with the omnibus (my first one is on its way) but I think the dedicated SBUS pin is
UART2 UART1. So, when you get to Betaflight, you will go to the "Ports" tab and for
UART2 UART1. you will select "SerialRX". After that, go to the "Configure" tab and look for the box that says "Receiver" and in that, choose Serial for Receiver Mode, and just below that, select "SBUS" for the protocol.
For some pictures of this, refer to this page -
https://oscarliang.com/sbus-smartport-telemetry-naze32/ The first two picture of the configurator show what I just explained.
Let us know if you need any more clarification.