What is your definition of a quant ? I know plenty of phd at my place end up as researchers i.e fundamental analysis, not technical analysis. If your technical skills are up to par then its possible to get in as quant and not being labeled as too theoritical.
There is not a clear boundary here.
I talk to people who have "quantitative analyst" on their business card whose work is essentially looking after a few buggy spreadsheets.
Nor is it constant over from one month to the next where "theoretical" quants can be debugging C++ or sigting through data, or puzzling out some hard maths.
Some quants aren't called that, instead labelled as "strategist", but of course many strats aren't quants. Senior quants often don't have this in the job title.
You need to look not only at the work they describe, but at your future bosses and peers to get a better feel for what work you are signing up for.
My personal opinion is that a PhD in Economics would be a big plus. I expect that most traders have a lack of understanding of economics and someone with that understanding could benefit a trading desk a lot. What do quants know about the IS LM curves anyway.