The HiBylLnk feature is really a smart remote control, part of HiByMusic. It can be configured either as a "controller" or a "server" depending on the device where HiByMusic is running. On a HiBy R2 or R3, HiByMusic is really the only app running, so HiByLink is found on the R3 settings and there is only one on/off toggle for a "server" mode. An iPhone running HiByMusic, also has only one HiByLink toggle, but it is for a "controller" mode. An Android phone or an Android-based DAP running HibyMusic have both "controller" and "server" options available, but it's an either/or--you can’t select both at the same time. HiByLink works on both WiFi and Bluetooth connections.
When you toggle HiByLink on your R3 (as a "server"), it allows a HiByLink "controller" (phone, or HyByMusic Android DAP) to operate your R3 remotely: pause/play, previous/next track, navigate and select tracks from the R3 library etc. The controller shows what is playing, including Album Art, but the tracks actually play on the R3. Not all the settings of the R3 are available on the controller. IMO, HiByLink works very well when I pocket the R2 or R3, with a wired HP or IEM, and I want to control what's playing locally from my phone through BT, not taking the R2 or R3 out of my pocket. For the cases where I want to cast something from my iPhone to the R2 or R3, I connect them as BT receivers (hotspot + Airplay is another option in my case).
HibyCast is a "remote desktop" feature, it is not part of HiByMusic (it's a separate app), it is not related to HiByLink, and it only works on Android for now (not available on R2/R3 or iPhone). It requires a WiFi connection. This is how it works: you start the HiByCast "server" app on an Android-based DAP, you then connect to it from an HiByCast "client" app on, for example, an Android phone. You then have total access to the DAP from your phone and can control it as if you were operating the touch screen directly. You can obviously control the DAP HiByMusic app, but any other app, browser, system settings etc. as well.
I don't know much about Android (I'm mostly on iPhone...), but I don't see HiByCast as being much different from other "remote desktop" solutions... unless I'm missing something! In my limited experience, it's a little slow, and not as convenient or reliable as HiByLink when you only want basic control of what's playing.