How to communicate with the advisor if someone wants to downgrade his PhD to Master in university A and then apply for PhD in university B?
Your wording here is perfectly fine for explaining your situation in university A. Your advisor is the one who works there and knows about internal dynamics of the university.

And also, name of the institution is important at a certain level. What values you as a scientist is the quality of your publications and thesis. Finishing a PhD in university B might be a challenge, but you can set your goal to make better publications than people who have PhD degree from B.

For the last point, there is always postdoc, which will probably be the most productive period of your life. You could finish your postdoc at university B and achieve your goals anyways.