It's not entirely impossible that there could be corruption on your system drive, but I don't think that's the likeliest option. If you want to completely remove Transmission and the plugin and you've installed the plugin from OMV then issuing apt-get autoremove --purge openmediavault-transmissionbt should normally be enough. That will purge the plugin and all it dependencies.
What I think you're seeing and the reason to why it worked before, but not now is a change with later versions of the plugin (or rather, a reverted change to how the original plugin actually worked). The plugin used to add the user debian-transmission to the users group (which means that it would've worked for you now since users has enough permissions). If you look at the NAS it's working on I would guess that issuing groups debian-transmission will show the users group. The issue with that is that system users shouldn't be added to the users group without the consent of the administrator since it can be a security issue (think of malfunctioning services having the access to all files and directories which the users group has access to).
So, what you can do is multiple things. You can either add debian-transmission to the users group (which I personally wouldn't do), or you can use ACLs to manage permissions. The con with the second option is that ACLs sometimes can be hard to work with. The third option (which I use) is to create a new group, for example download-adm and add the user debian-transmission to it. Then you change the group of the download directory to your new group instead and it should work. Now you can add all users that should be able to manage downloads to the new group and everything should work as you've intended. For read only access you can then set permissions to read only for others.
The permissions are still wrong from what I can see from your picture, so you need to figure out why they didn't change when you tried to change them.
I would've issued these commands:
A last suggestion would be to select the text in Putty that you want to show us and then right click it to copy it. Then use the code tag on the forum and paste the output in it. That way you don't have to upload pictures each time.