It doesn't mean the U.S. must protect anyone and everyone around the world who fancies themselves "Americans" ... it just means that extending U.S. interests to foreign areas would be legitimate if the purpose is to pursue the territorial expansion of the United States if the territory and Congress are on board with pursuing the admission of one or more new states.
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.