EDIT: He beat me to it.
Steven, please correct me if any of this is incorrect but I will try to explain this to the best of my knowledge.
Because you chose save to MusicBee only, it won't actually write anything to the tag, it will only keep it in the MusicBee cache.
If you click open the Tag Editor and click on the '...' next to artist, you are able to enter an artist and choose their role (Artist, Guest Artist, Performer or Remixer). Artist, Performer and Remixer will link to ID3 tags in the file, Guest Artist and Display Artist will not as they are currently defined by MusicBee only.
If you still want to see a separate field in the Tags (2) tab of the Tag Editor, here is what you need to do:
1. Go to Preferences, Tags (1). Click on Define New Tags...
2. Enter Remixer into the Identifer field and click Save. The other fields will automatically be populated.
3. Under the Custom Tags area of the Tags (1) preferences, enter the name Remixer as the display name of custom 1 (or any other custom tag number).
4. For 'save to music file as tag', select Remixer from the drop-down list.
5. Click 'Save'.
The Remixer tag should now show under the Tags (2) tab of the Tag Editor. It will link to the TXXX\REMIXER ID3 tag and you can check this in the Tag Inspector. However, the values from the one under the Artist Splitter and the one in the Tags (2) tab will not sync in my experience. I don't know if this is intended behavior or not.
I hope this explains it.