> They went to Oregon with Betty, the maid, and the cook.

Betty could be the maid. English meaning depends partly on word order:

  They went to Oregon with the maid, Betty, and the Cook.

Still ambiguous.

  They went to Oregon with the maid, the cook, and Betty
  They went to Oregon with the maid, the cook and Betty

In the former, I suppose the maid might be the cook also. The latter moves more easily and with less ambiguity.

If Betty was the maid, I would write…

     They went to Oregon with Betty the maid and the cook.

The maid implies there was just one. Who travels with their only maid? Who will keep the manor lights on?