Before every election in the US, there is a cutoff date by which one has to be registered to vote -- if you miss the cutoff, you can't vote in that area. People who are already registered to vote in California and move to another area may be able to vote provisionally -- I would have to look up the rules. But every step of the process has a similar cutoff date: e.g. in this June's primary, the deadline to apply to vote by mail had to be in the Registrar's office by 5:00 PM on May 31, 2016.
I would read that April 3, 1920 to mean that these are all the voters who have registered to vote or have submitted a change of address to the Registrar of voters through the end of that business day (to and including means it is inclusive of that date). This is likely to determine whether they could vote in an election sometime in June of the same year. Can you browse back to the beginning of the volume to find more information in the front matter?
If not, how else would we check this? If this were a contemporary question, we could look up the rules in the current California statutes online by reading and searching the Election Codes. So we need to find out what the rules were for April 1920, and to do this, we need to find the statutes that were in effect in April 1920. Michael Hait, CG, demonstrated how to do this in his recent webinar, “Finding and Using Online Legal Resources”, which was presented at GRIP. Unfortunately the link for California statutes online which was listed in his handout is no longer valid:
California State Assembly, Office of the Chief Clerk, Archive:
Searching on Google for the title in Hait's handout gave me a working page, California State Assembly Statutes, which is here:
(I'm leaving both links in the answer to give people more targets to search for in case the website gets redesigned again. Plugging any once-working link into the Wayback Machine at Archive.org would give an entry point, if you need to try and dig for it that way.)
Hait also demonstrated how to look for results in printed versions of the California Statutes via Google Books, the Internet Archive, or Hathi Trust.
If you don't have any luck plunging into and searching or browsing the 1920-era code, try the inchworm brute force method -- find the relevant section of today's code, then use the references to the law which was superseded to walk backwards through the historical statutes until you reach the period which is just before your target date. Another complication is that some statutes take effect as soon as they are passed, while others don't come into effect until a certain date, so if you have to pin things down very specifically, you have to watch out for that as well.
These registers will be prone to the same problems as other printed references -- people could move and not bother to give their new address to the Registrar of Voters, and so an old address might linger in the rolls. The rules about whether you can vote at your old precinct if you have moved recently, or whether you can use a provisional ballot at your new precinct should be outlined in the electoral code.
I haven't moved in quite a long time, but I vote regularly, so I am still on the rolls at the same address. However, we aren't required to show ID or prove our address -- we only tell the poll worker our address, and they have us sign to confirm we showed up at our precinct. If you don't vote regularly, they may purge you from the rolls -- again, the electoral code should say.
All this assumes, of course, that the poll workers and the registrar of voters did what they were supposed to do, which isn't a safe assumption.
I have a divorced couple in my research who appeared at the same address for years, which puzzled me. I thought that it was just one of those stale directory entries. Eventually I came to my senses and realized that the address was an apartment building -- it's also possible that the husband moved out and got another apartment in the same building.
I would start by looking up what kind of building 1131 Ocean Avenue is, to see if it is a single-family residence or a multi-family residence. If you can get at historical property tax information, that would tell you who owned the building and could yield other clues. Since the address you're looking for is post 1906, it could be a relatively recent dwelling.
Also don't neglect historical newspapers, which could tell you something about the neighborhood or have reminders and news articles about voter registration deadlines. The California Digital Newspaper Collection might be a good place to start. And don't forget historical maps!
Very important addendum:
Remember that this is 1920. Census Day in the US was January 1st, 1920. San Francisco's actual enumeration days should not be that far off from census day itself -- they are far less likely to be affected by weather than other parts of the country. But this census is the outlier when it comes to US Census Days, which otherwise are in April or June. As with any US Federal Census, it's a crapshoot whether the householders gave the information that was current as of census day or as of enumeration day, or that anyone from the household gave the enumerator the data. If you haven't looked into the nitty gritty of the US Federal Census already, see Who Talked to the Census Taker? by archivist Claire Prechtel-Kluskens for resources, and Michael Hait's US Federal Census Pathfinder.