Titles in our industry are such a mess. I really can't stand 'UI/UX Designer'. If I see that title then usually that's a signal for me that the hiring manager doesn't really know what they need.

To me product designer means someone who's a hybrid, an alrounder - basically a UX designer with following traits:

- good research and analytical skills

- decent or great visual design skills

- ability to create sophisticated prototypes (not just creating hot-mapped images)

- someone who has good technical knowledge, so they don't design something that's going to be really difficult and expensive to build

- someone who considers product and business matters, i.e. ability to strike the balance between user and business needs.

What are you hoping a PhD will give you?

I'm not sure where you are, but if you want to use your skills for something good, why not try to help your government - improving their online services etc.

See for example what the UK did to their government sites - a great investment that made it easier for millions of people to understand and use their online services.

One example: https://beta.nhs.uk/service-manual/design-principles/

edit: formatting issues