To answer my own post above, it looks like there are some caching features that I didn't fully understand. In particular:
1. It seems that, if an animated gif has already been displayed in a shiny app, the animation cannot be repeated (assuming the gif does not loop).
2. If you create a copy of the original animated gif (same file, different filename), shiny seems to recognize that the copy is identical to the original. When the animation displays for one of these files in a shiny app, the animation from both files cease to display thereafter (this is the caching part that I don't fully understand).
3. To get the same animation to display twice, I had to create two animated gifs, with one having a slightly smaller dimension than the other. Now, the file sizes differ and they are not recognized as identical.