research undergraduate - How to show a student the work done in the lab is the lab's (or is it?)

Difficult to title this, but in essence, I am working on the premise that ethically, the work done in the lab belongs to the PI and the lab as a group. So if there is another question that deals with this I have not found it.

In the case where a lab takes undergraduate interns or UROP positions, such that the undergraduate is not a student of the PI, how to inform them (as it is their first research experience) that the work they contribute to is not their own research to go and publish.

In a situation such that the student is put on a project and is guided to developing some code or doing some analysis (without their own substantial intellectual contribution, just development and applying standard methods), the student then leaves the internship or position. How to make it clear to the student without some threatening way, maybe by showing some international/online standards that explain, they can not take the work they did and publish it on their own?

Likewise, how should the PI reasonably decide authorship, and explain that students authorship or lack of, due to insignificant contributions of the final work.

For example, on a given project, I may have 4 or 5 undergraduates over time interning and doing some develop, exploration, etc. I want to be encouraging, and so I try to over emphasize the 'great job' they are doing. At the same time, I do not want them to misinterpret this as them being given authorship. In some case, I had an undergraduate that after awhile of working together, decided to drop everything and quit. The student now seems to have an attitude that the work they had been doing is theirs to go and publish on their own (in some low quality journal, as it is just part of a larger project that they do not have all material for).

Answer

I think that you are not referring to the right reason why your former students cannot simply go on and publish the small projects they did under your supervision in lower-tier journals (or any other journal for what matters).

It's not that the work done in the lab belongs to the PI and the lab as a group, it's that you and possibly other people in your group contributed to these projects, at least by providing scientific guidance and supervision (I think the answer to your question is to explain them that bit).

As such, your former students cannot claim that the work is solely theirs by not mentioning you as co-authors and, as a corollary, cannot publish without your explicit consent.