A table is in 1NF when every column has a fixed simple data-type that is understood by the database system, and every row has a primary key.
This is your own definition ?
If not, please indicate references.
For example, people may leave off the primary key requirement because they know that the base relations of relational database theory are required to have a primary key
In fact there is no primary key in relational theory. Or at least, that expression is obsolete (Date on database: writings 2000-2006, p117).
And there is no requierement.
By definition a relation has at least one key (a "candidate" key); it's not possible to have duplicate tuples.
Concerning the "A Table which is not in 1NF" part.
I'm not sure to understand. Are you really saying that 1NF violation is somewhat related to the application using the database ?
this is a claim made only by people who don't understand domain theory
I'm sure a lot of people (including me) don't understand that point. It's worth explaining it clearly.