Here is the link to the HP USB Dock DisplayLink Driver. I would assume either adapter would use this driver, but like I stated, there is no documentation for the non-elite version explaining this. So I am having to take a guess.
The solution marked as correct to your post only really referred to the user manual for the USB-C dock. This just tells you have to connect a monitor (so would suggest that two external monitors are not supported).
But the USB-C dock does have two video outs (DisplayPort and HDMI) so does that mean it could support two *seperate* external displays. And I understand that DisplayPort 1.2 and above supports 'daisy-chaining' monitors from a single DisplayPort output, so that may be another way of adding a second external monitor. But I can't see anywhere in the documentation any confirmation of what version of DisplayPort is used in the dock.
But maybe you got it to work with two external monitors? This is what I'd like to do, so please let me know whether you managed!
I'm running two external monitors - no magic required. I didn't even have to open the DisplayPort software. One is connected VGA, the other HDMI. In Windows display settings I have the tablet display disabled, the external displays set to Extend Desktop. Thats all I needed to do.