This morning, I was re-reading the wonderful article Full Life, Empty Pockets, by Harriette Jessup Proctor, which appears in Issue Number Two of the Santa Cruz County History Journal. The following line caught my eye: "...Alexander's, an excellent Italian restaurant, in Billings' big Victorian near the highway." Bingo!
So it turns out the caption accompanying the photo in the SCPL collection is correct (sort-of). I'll attempt to track down the exact location and the identity of the original owner, but I'm guessing the Billings house stood where the Pasatiempo Inn is today. Of course, the area (part of Rancho la Carbonera) wasn't called Pasatiempo when the house was built, sometime before 1900. Marion Hollins invented that name in the 1920s. That was my mistake - when I read the photo caption, I assumed (and we all know what "assume" is an acronym for) that the house was claimed to be part of Hollins' Pasatiempo development.