Question: I have so far only worked with WS2812 strips and understood, that the WS2813 have a backup data line. But how do I connect the first LED to my Arduino (acts as a controller)? 

Many tutorials said I should just wire the BI to V0, but doesn't this eliminate the backup functionality? So I thought about wiring both data pins to my data source? Is this a good idea?

Answer: 
No, not connecting the BIN doesn't eliminate the backup functionality. As for what to do with it, the official datasheet at <http://www.world-semi.com/DownLoadFile/117> doesn't say, so I won't speculate.

To show that the backup remains consider that the BIN of a WS2813 LED is connected to the DIN of the previous LED.
![Typical application circuit of WS2813B](https://i.imgur.com/ucVLOx2.png)

If the first LED is working, then it receives and shows the data on it's DIN and then forwards the next pixel data via DOUT to the DIN of the next LED in the chain. Note that the second LED will now receive it's data on both BIN (direct from the first LED's DIN) and on DIN (repeated by the first LED).

If the first LED is non-functional, then the next LED first sees the first pixels' data on it's BIN (and disregards it). It is then expecting data for itself on the DIN channel but receives none. Thus it goes into backup mode and uses the data that came in on the BIN channel direct from the DIN of the first LED.

Thus, DIN is the one that needs to be connected on the first LED.
Based on the documentation and experimentation: do not connect them together because they are not interpreted the same way. D is the data signal and interpreted as such and B is expected to be the data signal with address shifted -1 so the addressing stays in sync. Do not connect your controller to BI. Do connect the BO at the end of one strip to the BI of the next strip in your daisy chain. If you are not chaining them together, you do not need to connect anything to BI.