How have you presented your sites address?? If it is presented as California, you are likely experiencing a two-fold effect whereas the term California is explicit and well understood as well as an exact match. Semantics is not needed.
Keep in mind that the accepted abbreviation for California is CA not cal and not calif. It is not terribly likely that semantics will pick up strong signals for these alternate abbreviations (cal and not calif) even if they seem to be well known. It may be common to use these abbreviations in California, however, are they likely to be used in New York or Florida? What about London England or Stuttgart Germany? Perhaps not. As far as CA is concerned, you have a problem in that it also stands for Canada, Certificate of Authority, Chartered Accountant, Circa, Court of Appeals, etc. Again a semantics problem. It is very likely that a stronger signal for CA exists for California and Canada and not for Court of Appeals for example. This is a given of course. However, semantics is situational.
There is likely a stronger semantics signal for Manhattan Beach and California than CA. I would expect that CA should match just as well under the right circumstances. If I type Manhattan Beach in Google's search textbox, I get-
Please note that your Manhattan Beach is not the only one.
But if I just add a space, I get-
Now I type your search and I get-
This tells me that California and CA have strong semantic signals.
Not knowing your site and how you have tried to weigh search for your site, I would recommend using Schema.org mark-up for your site at least in the About or Contact page and possibly if not in those two places, in the sites footer (which may be better). I would use the state abbreviation- CA since that seems to be what is expected (double-check). I would let semantics and the strong signals for CA do the work since we know it exists.
As for the other abbreviations, I would discard the notion that these are valid uses. Without posting more images, I found that these are not recognized terms as far as Google is concerned. I ask, how many people are really using these abbreviations to find your site? Not many I imagine. Darwin has a way of sorting this stuff out. Google does too.
Follow my advice on using mark-up and do not worry about too much. It will take some time for the change to come about- expect a couple of months. As well, back links are another strong factor that will help you in the SERPs. Also consider citations such as in local ads, local newspaper articles, and mentions in blog posts (you do not actually have to write a blog post, just be mentioned) just to name a few opportunities. Citations are a powerful tool often overlooked. Site age is another strong factor as well as good SEO. I discuss the basics here a lot so feel free to use the SEO tag and poke around. As these things develop, you will find that you will begin to bubble to the top of the SERPs. But it can take one or more years to become a top performer. Less for a nice niche like yours.
