Berkeley HS, Piedmont HS, Albany HS or Alameda High

Piedmont HS is very good but it is not diverse. It is a predominantly white with a substantial Asian population and there isn’t much economic diversity, if that matters to you. Our friends who are 6 figure salary earning professionals often joke that they are the paupers of Piedmont. 
 

Berkeley, Albany, Alameda HS all are well regarded and Berkeley and Alameda are more diverse. Houses in Albany tend to be rather small. 

I live in Oakland zoned for Oakland Tech HS and we appreciate many things that our area offers. We will consider Oakland Tech HS  when the time comes but we do want to keep private or charter schools as an option or Bishop O’Dowd. 

I think Albany High would be tough socially to break into since it's a small community and many of the kids are acquaitned since either K or 6th grade.

Of these four, Berkeley is your best bet for diversity--Albany and Alameda Highs are strong academically but both predominantly white/Asian and middle/upper income. Piedmont is predominantly white and upper income. (Encinal High in West Alameda is more diverse both ethnically and socioeconomically and could be worth exploring.) Given the neighborhoods you like, I might look at areas zoned to Willard Middle/Berkeley High. Berkeley High is very large but it does reflect a big cross-section of the community and most kids we've known who've attended have found their circles. Good luck with the search!

BHS is diverse and has lots of clubs and opportunities to make friends, but might be a little overwhelming for a shy student as it's a big place. I know a number of kids who went to Oakland Tech which you didn't mention, but is ranked well, academically strong, and very diverse.

With that said, you should note that the Great Schools score (assuming that's what you are looking at) is impacted by ESL and lower-resourced students. So if you truly care about diversity, you might want to be skeptical about that score. We moved from an Alameda 9/10 scoring public elementary school to a more affordable bay area suburb with a 2/10(!) middle school and a 4/10 Title 1 high school. The experience has naturally had ups and downs, but overall it has been positive and we don't regret it. You'll find good/bad teachers and kind/mean students in every school. Almost every high school will have a more rigorous track if your student is academically strong.

This is an old blog post, but it helped lift the fog for me as we were fretting a bit about moving to a lower-performing school district, and I think it's still relevant:  https://designmom.com/why-we-dont-stress-out-about-choosing-a-school/

One thing that was a marked difference was how much the Alameda elementary school we moved from was driven by getting the students to score well on standardized tests - constant communication about it, prepping, and candy being given to the kids on test day(!) There's little to no emphasis on standardized testing at her current high school nor will they be hosting a PSAT. If she decides she wants to take the SAT/ACT, we'll need to seek external prepping.

Another thing I find interesting is the percentage of students a school sends to the UCs. Although a high school doesn't rank well, and might not have as many 4-year college-bound students right after graduation, percentage-wise they will probably have as many or more students accepted to UCs (or even Cal) compared to other higher-ranked high schools: https://www.universityofcalifornia.edu/about-us/information-center/admi…

I'm not sure if your omission of Oakland Tech means you've already ruled that out, but it's kind of an "it" school right now, and deservedly so. Our college sophomore and high school sophomores both had/are having excellent experiences there. The Paideia program is incredible, as are many of the academies (especially engineering) as well as the drama program. So many of the teachers have been the type that truly care and go way above and beyond, and the principal is very dialed in with the kids and doing a great job with campus culture. It's also about as real and diverse (both ethnically and economically) as it gets. It warms my heart every time I'm on campus. Oakland schools get a bad rap; Oakland Tech is amazing.

Berkeley High is a good place for strong students interested in clubs/activities. My student (white) had a diverse friend group. They also did some activities in the community— Berkeley Rep, ATDP some summers, etc. They did very well on the standardized tests without any paid tutors, and were accepted by a highly competitive college (without private counseling). Your mileage may vary — my child was not particularly competitive or outgoing but they were very self-directed. 

My daughter went through Alameda High and my son is there now. It's not as diverse as BHS or Oakland Tech, primarily in that it has a tiny Black population. But it's a lot more diverse than some of the "fake" (to use your daughters' term) suburbs. AHS is more Asian than white, and has a lot of diversity within that - recent immigrants vs. many generations in the US, well-off and not, many countries of origin (China, Mongolia, Philippines, and on). Encinal High is on the West End and is as diverse as it gets - when I looked a few years ago it was 20-30% each white, Latino, Asian, and Black. Both are really strong schools and have a lot of community support, AHS is higher income and tests higher. AHS is open campus and when my elderly parents used to come over and eat lunch on Park St. they'd often remark how nice the AHS kids were. At the risk of hyperbole, Alameda has been an idyllic place to raise our kids. They've walked to school K-12, much of the island is within walking distance of Webster or Park St for kids to hang out, and the beach and the bay are just down the street.