What is the typical PhD application procedure in the UK?

I am nearly finished with my master in mathematical finance. I have finished my thesis and have one course left to take. I am finishing it in a third world country and am interested in pursuing a PhD in stochastic analysis in the UK.

I have heard that the application process for PhDs, unlike applications for masters or bachelors, involves first speaking to faculty and then to the school. Is that right? That is, when I apply I should already have a professor in the university willing to be my doctoral advisor?

The exact procedure I heard is something like:

1. Read up various literature on your desired dissertation topic which should include several textbooks and even more for recently published articles. (If necessary, study for IELTS, GRE, etc.)


2. Come up with a PhD proposal.


3. Contact relevant faculty of the universities to which you intend to apply.


4. Discuss your proposal with them if they're up to it.


5. One of the following: Revise proposal if needed, completely change proposal or get referred to different faculty member.


6. Actually apply to the university.

So, is that how it actually is?