Looks like an orchard apple to me. Maybe it's an orchard apple with small fruits, making it a crabapple on the basis of fruit size only. Anyway, you are lopping the top back too hard, causing rootstock sprouts to grow even more readily. Prune these more rigorously and the top less so.

And cut off that dead twig. Study fruit tree pruning guides in print and on the internet to get a better idea of how to manage this specimen. Being in California there should be plenty of locally generated information available from California Cooperative Extension, fruit husbandry is a big deal in your state.

Okay, thanks so much for that information! We moved into this house about a year ago and this is how it's looked ever since we moved in. I was scared to touch it. I'll follow your advice and get ahold of the CA Co-op Extension. Thanks again.