There’s my problem. I’m just looking for a definition and method to identify an interest. You have taken it one step too far by creating and stating an interest.
The place I am going with this is pretty simple. If we can agree on the definition of an interest, it allows us to have an argument about what is and what is not an interest. Using your #1 definition (protects the life, liberty and property of Americans;) in your original post works. You can argue about the things in your #3.
I’m not saying #1 is complete or incomplete and I’m not saying #3 should or shouldn’t be an interest.
You have taken it one step too far by creating and stating an interest.
The Constitution lays our a process for admitting new states to the Union, so I just assumed that they implicitly established a national interest in expanding the country under the right conditions.