Should I cite myself in my PhD proposal?

Yes, you should cite all relevant research: When conducting research projects that extend over multiple papers it is not unusual to want to refer to your past published papers on the topic. These should be treated just as with other relevant literature ---i.e., if they are relevant then they should be cited properly.

In general you cite yourself just as you would any other person. Failing to do so, while repeating earlier things you have published, leads to a charge of self-plagiarism. It is an odd concept, but is treated seriously.

Ordinary plagiarism is when you claim the work of another as your own. Self plagiarism is a bit different, of course.

But avoiding both types of plagiarism via proper citation has the purpose of placing a work of scholarship within its complete context. People reading a new paper want to know what it is based on, whether by that author or another. Having the citation lets a reader go back to that context (and possible further citations).

So, while this isn't an actual publication you are developing, the same rules should apply. The reader wants/needs the context.

As others suggest, you can and should cite yourself. But, more importantly, you should clarify this - not via the citation but in the proposal text itself. That is:

• Explain what kind of research you were doing before, what you motivation was, what limited your scope (if anything) etc.
• Explain how your proposed Ph.D. research continues your previous work - as such, i.e. don't just say "It was established in [3] that bars can be frobnicated; I propose to extend this result to baz" - where [3] is your own paper.
• Explain how your proposed Ph.D. research goes in a different direction / does not continue your previous work.