Here's what has happened [stupidity disclaimer]
:
- I downloaded the originally transparent image and tested with it in Chrome, Safari and iPhone.
- The image was 1MB so I downsized it to PNG8 (and forgot to allow transparency when exporting).
- The cached version of the image was used in the browsers I had been testing with (which I didn't notice).
- When I then moved onto Firefox, the new non-transparent was being used (because of no previous caching).
Result: Firefox renders transparent PNGs fine - happy with this question being closed as "too localized".