WhiteWalker

The two options would lead to extremely different outcomes. If you enter into UCLA's Applied Econ program, most likely you will never be admitted into any decent econ phd program (let's say top 30) as the preparation is just inadequate. You might be able to get into business phd programs though, if that's what you could accept. In terms of job prospects, of course the UCLA brand is good enough to send you into decent industry jobs. On the contrary, MA at UT Austin provides rigorous training and the phd placement outcome, as far as I know, is great. However, industrywise it doesn't provide as much value added as UCLA's MAE, at least in regions outside Texas. Hence choose the program based on your career aspirations.

It really depends on the sort of topics you wanna work on during your doctoral studies. Political Science schools have the subject of politics as their foundation and the methodologies applied will be very different from what you would expect in econ programs. For policy schools, training would be more school-specific (at Harvard Kennedy for instance, you could receive training comparable to that of a std top 20 econ shop), but one thing for sure is that they will put a strong emphasis on policy research. Plus, from what I heard, it would be a hard sell for you on the econ market as the name Public Policy puts you at a disadvantage. In a nutshell, if OP is good with the thought of joining a policy school or a mid- to lower-tier econ shop upon graduation, a public policy PhD is perfectly fine. Otherwise better do a std econ program and, if necessary indeed, contact the ppl whom you wanna work with outside the econ department. Hope it helps with your question.