I didn't say they were the same thing. And I meant what I said: zoning allows a shed without building permits in most places. It depends on the jurisdiction but because of model zoning codes the rules are almost all the same across CA.

In CA and most of the US there is usually no separate "zoning permit", although it exists in a few places (I see berkeley and sonoma county). Usually zoning and the development and land use codes just say what you're allowed to do, and are enforced based on helpful neighbor-driven complaints. But you usually don't need a permit up front except for things like conditional use and special use permits, certainly not a shed. For things requiring a building permit, the permitted use (zoning) is just part of that process.

The OP can and should check the city/county and find out. The point I'm trying to make is there is actually a fairly high likely hood that even in over-regulated CA you can build a 120 sq. ft. "shed" or "playhouse" in the backyard without a building or zoning permit. Without knowing the exact jurisdiction, this is just general advice about the most usual practices - for example San Diego, LA, San Bernardino cities and counties.

One thing I should mention: technically plumbing/electrical in the shed or calling it or using it as an "office" might not be kosher, but in practice everyone does this with no issue. It's just a very nice "playhouse".