[Galaxy S III I9305 (4G LTE + 2GB RAM) Q&A, Help & ] Oswald Boelcke:
Update IRT the six posts above...

BLUF: I ended up with reverting to SuperSU v2.76!

Flashing of PHH's SuperUser.zip worked without any problems in both ways i.e. applying either "full unroot" or "switch superuser" in the still installed SuperSU v2.78 prior to booting into recovery. Even after "full unroot", the SuperUser.zip clearly re-roots the device as [MENTION=7018227]limjh16[/MENTION] already indicated. I also realised that at least I had to install the SuperUser-Interface-APK after flashing of the SuperUser.zip. When I installed the APK while SuperSU was still installed, when opening SuperSU to either apply "full unroot" or "switch superuser" I received a warning about two superuser installed (however, the second was only the GUI) and a question if I wanted to delete the second one. I denied the latter. Despite that, after flashing SuperUser.zip and restart, the SuperUser-Interface-APK wasn't available on the system.

PHH's SuperUser provided me full root management; however, one annoyance and one problem that definitely dragged me away from PHH's SuperUser and back into SuperSU but reverting to v2.76.

Annoying only was the permanent notification when opening Titanium Backup about a superuser binary that might create problems. Even if I ticked "Don't show again" the message came up every time I opened TB.

My severe problem was that I was positively connect to data after restart but never achieved an internet connection - neither on WiFi nor via mobile data. I'm convinced this wasn't an issue of SuperUser but of my IPTables. All my root apps requested root access and got it granted by SuperUser except one: AFWall+! For whatever reason, AFWall+ did not request root while I was on SuperUser i.e. I was unable to modify the IPTables and was stuck with no internet connection. After conversion to SuperSU v2.76, all was back to normal i.e. root access for AFWall+, too.

Some off topic remarks:

On XDA there aren't anymore any links leading to Chainfire's SuperSU v2.76 - at least I didn't find any. Everything is either leading now to the above mentioned new SuperSU v2.78 thread of CCMT or to the download page for v2.78 (seems to be still on Chainfire - at least his name appears in the URL but probably sold, too). Only a web search brought me to the following download page for v2.76: https://download.chainfire.eu/969/SuperSU/.

Another curiosity - While the dev's name in the SuperSU Prokey APK has been modified in Google Play Store and doesn't go under Chainfire's name anylonger, the SuperSU v2.78 APK itself hasn't, and Chainfire is still mentioned as the owner. I don't want to call it fraud but certainly at least deception. Why is Google able to change names in the Prokey but not in the connected app? Only comments in the play store like the grateful one of [MENTION=7125933]Sam Nakamura[/MENTION] makes it visible to other users that Chainfire isn't anymore in it.

Edit (2016-09-18): Next time, I'm going to flash my current ROM (e.g. for a revised version) I'm giving PHH's SuperUser another try.

Edit (2016-09-19): PHH's SuperUser incl. the former troublemaker AFWall+ (Beta) now succesfully working!

Dirty flashed in this sequence: ROM, GApps pico, AGNi, Xposed, SuperUser.zip; SuperUser-APK was already installed, as I had flash superuser.zip and installed the APK even prior to flashing the ROM but at that point AFWall+ didn't asked for root. Therefore uninstalled AFWall+, fully un-rooted the device by SuperSU v2.76, and then commenced the above flashing sequence.

Two remaining issues:

AGNi Control requested and got root granted by SuperUser; however, when opening AGNi Control none of the tweaks is visible. I'm going to post that issue in AGNi thread. Checked also with AGNi v5.10.3; same behaviour as for AGNi v4.5.5.

Edit (2016-09-25): Due to the issues with AGNi Control - and especially due to "fuel" management, I need to use AGNi instead of any other kernel, I reverted to Chainfire's SuperSU v2.76. At the AGNi thread, I didn't receive any reply in regard to this issue.

Edit (2016-09-25): Root Essentials is still stating in its module "Root Checker" that SuperSU is the root manager. All other applications state SuperUser by PHH to be the root manager. Going to post Learnt at [APP][3.0+] Root Essentials that the application actually doesn't indicates the root management tool used but the mentioned tile is only a link to Google Play Store to allow download of SuperSU i.e. from my point of view it's an advertisement for SuperSU v2.78 by CCMT.