Is 15/32" plywood thick enough to be used as a subfloor under tile combined with a 1/4" hardiebacker board?
The joists underneath are 2x8s that are 16" apart on center and only 4' long.
Most tile and mortar manufacturers want 1-1/4" (nominal) even with polymer-modified mortar. While your joists are unlikely to flex, the subfloor might. I'd use 1/2" cement board if height isn't a critical concern.
Alternatively, block between your joists on 16" or 19.2" centers. Even vertically-oriented 2x4s would do if fastened well.
From a practical standpoint, you want to use a “subfloor” during the CONSTRUCTION PHASE. (The subfloor will get dirty, scratched, etc. during rough framing and construction.) Then during the FINISHING PHASE install an “underlayment” for the installation of the tile. (The underlayment is actually a part of the FINISHING PHASE.)
Follow the Tile Council of America (TCA) recommendation for type of project (purpose of project) i.e.: waterproof, high traffic, etc. Then, the TCA will recommend the installation method you should use, (i.e.: thin set, mortar, etc. )
Then after you’ve selected the type of install recommended, you can select the type of tile, (i.e.: concrete, spanish, (clay), porcelain, etc.).
Then, you can select the manufacturer. The manufacturer will recommend: 1) subfloor, 2) underlayment, 3) setting bed, and 4) grout. Make sure you follow the manufacturer’s recommendation or you’ll void the warranty.
However, remember: subfloor during construction and underlayment during finish phase (installation of tile).
In order to answer your question, we’d need to know span of subfloor and purpose of tile floor. (However, most manufacturers will require plywood underlayment, not particleboard, OSB board, etc.) Generally, in residential construction, 1/2” AC plywood underlayment on 3/4” plywood subfloor on joists at 16” o.c. is acceptable, unless you are going to sit a large refrigerator on it...