Severe Voltage Drop When Applying a Load to Opto-Isolator Circuit

The input must not affect the output, it's a closed loop system, which means that in wide range of input voltage, it will regulate the output to what you configured.

And since i haven't seen such unregulated DC/DC for a long time, i suggest to first understand how do you connect it and what are you measuring. Specifically, you have to measure on the module itself to ensure regulation. The wires must be thick enough to hold charge current. Of course, in the end the measurement should be taken near the phone. If the cable is poor, you may need to raise the DC/DC output voltage.

Your two configurations are equivalent in output drive, although obviously the output logic is inverted between them.

The decision therefore comes down to which polarity is more fail-safe and idle current issues. Config 1 fails to high and takes active current to be low, with config 2 the opposite. Use whatever works best for your system.

One reason people knee-jerk to config 1 is that a switch between a line and ground with a pullup is more common. It is often convenient to be able to ground one side of the switch. Far more things come with built-in pullup resistors, as apposed to built-in pulldown resistors. If this signal were going into a microcontroller, for example, then R1 could be internal to the micro and you'd use config 1. In that case logic polarity doesn't matter since micros can generally deal with low or high equivalently.