How to write formal letter for PhD after completion of MS
If you studied in the same department, you (should) have at least two advantages:

1) You know the culture there -- how important "rules" are.

2) You know people: Professors (your MS thesis advisor), Master students, PhD students, your student union.

So try to use this to find out how to write the letter. Ask the people you know how they wrote their letter/ what they recommend/ if it really matters. (You may find out that nobody even cares if you really write the letter.)

In my university for example, there is also such a rule -- however, in reality, nobody looks at the letter and it's even possible to just write "I want to do my PhD". The official rule of course doesn't say so and secetaries also have no idea.

If this fails and you do not get any information, include at least: your name, your previous studies, your prospective advisor, if you will get funding and how and a detailed description of your project.

I wish you good luck!