Patrick, I'm a bit new at Server too, so this is just a guess: 'Allow' 'Write/Web Save' (third one down). That might give your user full editing privileges. Also, I'm not sure I'd use a mis-matched Desktop/Server (8.1 & 8.2), I suspect that's just asking for trouble.
Let us know if this works.
Looks as if the key was setting "Web Edit" to Allowed. The end users can now interact with the dimensions/measures via their web browser