Screensharing is not enabled becuase Remote Management is on.
Remote Management is intended for the ARD (Apple Remote Desktop) paid software. ARD used to manage a large number of Mac, such as a classroom of Mac.
Screen Sharing is used for individual remote access.
Now if you are using ARD as well as allowing individual users to Screen Share, then Sharing -> Remote Management would be correct, else Sharing -> Screen Sharing
Having said all that, unless you have used Sharing -> Remote Management -> Options to disable a bunch of remote features, then it should not affect your ability to initiate a Screen Sharing session.
But... You should run an Experiment with a system that is not allowing Remote Screen Sharing to see if Sharing -> Screen Sharing behaves different from Sharing -> Remote Management. I do not expect the experiment to change anything, but you should not leave any stones unturned 🙂
As long as the user account using the local monitor, keyboard, mouse/trackpad is the same user attempting to initiate a Screen Sharing session from another location. NOTE: I do not know if it would refuse a different user trying to connect, but eliminating that is just one less thing to worry about.
Now topic. Is the remote user really using a VNC client, or are they using the Mac OS X Screen Sharing client on another Mac? If a VNC client, what is the client, and have you given Sharing -> Remote Management -> Computer Settings a VNC password.
Now I suspect you are using Mac OS X Screen Sharing client, as the error message you posted, says "Please verify you have entered the correct name and password", and if using a VNC client, there would be not "name" involved. But I had to ask.
I do not think I'm making any good suggestions, as I have never connected a Mac to Microsoft's Active Domain, nor managed an Active Domain.
Is Sharing -> Remote Management -> Allow access for: All users enabled? If not, what is enabled?
If All users is enabled, then as an Experiment, try configuring 1 Mac specifying "One these users:" putting in the user that is going to connect to this experiment.
Basically I'm just trying to see anything that might shake things up. If something works, or gives different results that might shed light on what is happening.
You are posting in the Snow Leopard 10.6 forum. Are all the Macs running Snow Leopard? Or are there other versions of Mac OS X involved? Is there different behavior for different Mac OS X versions?
Assuming the Mac OS X Screen Sharing client is being used to make the connection, what version of Mac OS X is being used?
I do know that I have had issues using Mavericks (10.9) Screen Sharing client to connect to a Snow Leopard (10.6) system. So I'm wondering what mix of Mac OS X versions are being used. And if you have tried same Mac OS X versions for client and server so your Screen Sharing client and server are matched. NOTE: mixed versions should work, but I'm just trying to establish a base line with the least amount of differences.