OK lets try this...

Open your image that has a transparent background and only ONE layer!! You can not have a background layer along with your image layer!!
Then click on Save AS, Options>Select Noninterlaced and Optimized Palette then click on the Run Optimizer button.
Colors Tab: 16.7 Million colors
Transparency Tab: Alpha channel transparency & Exsisting image or layer transparency
Gamma Tab: 45455
Format Tab: Non-Interlaced
See if you have those types of setting in PSP8, I am running PSPX and never used 8!!
Ok I saved your test image to my desktop and opened it in PSPX and it opens with a transparent background!! So it is something to do with a setting on your PSP viewing ability....hhhhmmmm again!