Edit: I misread the question and interpreted it as being about including personal aspects in the usual statement of purpose, rather than writing a separate personal statement. Berkeley is unusual among universities in having standardized university-wide on two essays for graduate admissions, a traditional statement of purpose and also a personal statement, which often deals with diversity and inclusion issues. The advice I give below is aimed at other universities with a single statement of purpose, while Berkeley may be looking for something different in a personal statement.
Don't worry about including all these things in your statement. You don't need them all, and trying to cram them all in will very likely lead to a worse statement. The goal is to write something coherent and well organized, rather than a brain dump of everything that might be relevant.
Overcoming barriers is worth mentioning, but it should not be the focus of your statement. The purpose is just to put your accomplishments in the context of the opportunities available to you, so the admissions committee can judge your potential even if you haven't had the same opportunities as other applicants.
Academic service regarding equitable access could also be worth mentioning, but in most cases it will be at best a very minor factor. It could in principle end up as a tie breaker between two candidates, but it's not going to overcome any weaknesses in your application. Mention it if you have something impressive to say, but don't give it a disproportionate amount of space.
Understanding the barriers faced by others is generally not relevant. It's a commendable personal trait, but not one of the admissions criteria for graduate school.
Evidence of your research focusing on underserved populations or related issues of inequality
This is in an entirely different category from the other topics you mention. The motivation for and impact (potential or past) of your research are of real importance, and you certainly need to make this clear in your statement of purpose if you have something noteworthy to say.
And more importantly, how your personal background and experiences inform your decision to pursue a graduate degree.
If you are talking about telling autobiographical stories to explain how you arrived at this point, it's almost never a good idea. Discussing your academic experiences in college is worthwhile. For example, you might discuss what you learned from different projects (summer research, senior thesis, etc.) and how your plans have taken that into account. Talking about high school is generally a bad idea: it looks bad if after college the most relevant things you can think of to discuss are from high school. Don't even bother mentioning anything earlier. Nobody cares at all who inspired you as a child or what your childhood dreams or accomplishments were.
And is there any different between this statement and the SOP for undergraduate applications?
Yes, it's almost completely different, which means advice for undergraduate application essays is at best useless and at worst actively harmful for graduate applications. At least in the U.S., undergraduate admission often takes into account what graduate admissions committees would consider to be ridiculous fluff (e.g., Caltech asks applicants about "an unusual way in which you have fun"). Your statement of purpose for graduate school should ideally not contain any fluff, and if it includes more than a tiny amount then it will hurt your chances.