For polygons in polygons I would suggest using either within or have their center in. Within will only select precincts that are within the county. Have their center in works well if you have messy polygon edges that overlap a lot, however, if you have extremely irregular polygons whos centroid may fall outside the polygon boundaries. Just something to keep in mind if there is a ton of crazy elongated weirdly shaped districts.
Here is a link to spatial relationships that may help you understand your options better.
Also you will need to keep the Join Operation as Join_one_to_one as that will join and aggregate for you. To combine your fields you probably need to add a new field. to aggregate your data. Right click your votes field and in the merge rules select sum. This will sum the values of all the precincts that are within your counties.
help doc for the spatial tool:
help for the merge portion:
You can also use the Join Data Tool. I find it is generally a little easier to get your desired output, unless of course you need the full control of the spatial join tool. Right click the target features in the table of contents (California Counties) and select joins and relates and select join. The Join data dialog Box will come up. In the What do you want to join to this layer dialog box, choose Join Data from another layer based on location. Theen you can specify what attributes you want summed or if you want one to one, etc. The Join Data dialog tends to filter out all the options that are invalid for your relationship so they may make it easier to digest.