Linear Technology PMIC (LTC3566) - powering circuit entirely from VOUT1
Please look at the block diagram on page 11 of the datasheet. You'll see that there are two separate sections.


System that takes power in from Vbus and outputs to Vout and to/from the battery.
Buck/boost taking power from Vin1 and outputting to Vout1.


In the sample schematic you copied, notice that Vout is connected to Vin1. Therefore, Vou1 is downstream from Vout and what was said about giving priority to Vout also applies to Vou1.