Each of your examples is a title or a name. It is very common, particularly in the United States, to drop the definite article before nouns serving as titles. Each of these sentences (except for the one with "Mom", because it's used as a name) would also be grammatical had you included the definite article, so if you are unsure, it is safer not to drop it.
My brother is chief of this expedition.
You should discuss it with Mary, dean of our faculty.
He was appointed captain of the S.S. Minsk.
I miss Mom!
He was elected President of the United States.
You cited the last sentence as an example of one requiring the definite article, but it is quite common to omit "the" here as well. Also note that "President" should be capitalized when it is part of the proper noun phrase "President of the United States."
"Mom" should also be capitalized when it is used as a name/title referring to the speakers mother but not when it is used as a general noun.
"I called Mom." vs "I called my mom."
"When is Mom coming home?" vs "When is your mom coming home?"
The definite article should be used with driver, nanny, and butler, except when the words are used in place of a name or as part of a name. Examples include:
Driver, take me to Park Avenue.
Nanny Taylor, can I have more ice cream?
However, something like "I was just hired as driver for Uber" would be wrong. Unfortunately, there isn't really a formal rule to determine whether nouns can be considered titles/names. However, a good rule of thumb might be that more common nouns are less likely to be used as titles. Again, there is nothing wrong with something like "He is the chief of this expedition", so you can decide to keep the definite article when you are unsure.
