[goodoldneon]

They went to Oregon with Betty, a maid, and a cook.

Is Betty a maid or did they go with 3 people?

[happytoexplain]

This could be two people, but would normally be written with a different separator: "Betty, a maid; and a cook" (just removing the comma doesn't help because then Betty could be a maid and a cook). As-is, the implication is that this is three people. If you would like to make that more explicit, you would instead re-structure the sentence†, so it's not highly relevant to the serial-comma-vs-no issue.

†For example:

Betty, one maid, and one cook.

Betty, and a maid and a cook (a little awkward)

A maid, a cook, and Betty (depends on how you want Betty's inclusion to land for the reader)

[goodoldneon]

Right, you can change punctuation to clarify it. However, it doesn't change the fact that the Oxford comma could make the list readable as a parenthetical phrase.

I'm not saying the Oxford comma is bad. I'm just saying that it isn't 100% perfect as many people imply.

[meowster]

Three people; otherwise it should be, "They went to Oregon with a cook, and Betty, a maid."

Or better yet: "They went to Oregon with a cook, and maid named Betty."