Who should be author in an application note Work leading to the application note

During the c(o)urse of my PhD studies, I have written some analysis tools. I later collaborated with another PhD student to developed a proper analysis application. We then wrote an application note, to publish it in an journal. A third PhD student provided us with data, did an analysis using out tool and wrote the corresponding methods section in the application note.

The problem with authorship

By the rules of the journal, there should be three persons as authors: us three PhD students (Contributed work, and involved in draft). But as it is custom, is proposed an author list including us PhD student, the respective supervisors and the head of our research group/PI

However, my supervisor wants me to only include two PhDs and himself as last author. It seems quite important to him that the author list is short. Additionally, he wants to submit the paper himself to the journal. He has little knowledge about our work, nor does he understand the reasoning behind the analysis performed.

His sole contribution was to review the draft, so that all technical passages are now very understandable, as we explain our work to a scientific layman.

What I am asking myself currently


This is my first publication and I wonder if the demands oft my supervisor (him submitting to the journal, claiming to be the only "guest author") are justified, given the nature of his contribution?
Given this context, who should be author?


On a side note: While writing this, I also realize that the relation to my direct supervisor is quite toxic...

Edit: 

We three PhD students are not in the same Departement.
Authorship customs tend to be very field-specific, but I've never heard of discarding a co-author from a paper just for the sake of having a shorter list of authors. This sounds ridiculous: following this reasoning your supervisor could as well decide to be the sole author of the paper!

It's common in many fields to add a supervisor as co-author (usually the last one), even if their contribution is very limited indeed. However good authorship practice recommend that every co-author should have played a significant role in the paper. So in theory you are right: if anybody should be removed from co-authorship, it's the supervisor himself.

Needless to say, it's completely unethical to deprive an actual co-author of the credit.

If there is any room for argument with this strange request, I suggest the following: check a few recent issues of this journal and count for each paper the number of authors. You should be able to easily prove that papers with more than three authors are common and there's no reason to discard anybody.