Great questions, Diana. Important issues here.
The Link Info area of the Links panel shows 329 x 295 because you probably scaled it disproporitionally (squished it wider or narrower). The pixels in the 96 ppi image got smaller when you scaled it down, so the final (“effective”) resolution is high enough for print. At this point, you do not need to go back to Photoshop to resize/resample it.
(If it were a really high-quality photographic image, you might want to, and then apply appropriate sharpening, etc, but that's a whole other set of criteria. For most images, you don't need to and don't want to take that step.)
If you were going to place the 96 ppi image into InDesign and not scale it down, then it might be too low res, and would look jaggy. There isn't much you can do about that. I would still not open it in Photoshop and resample it, as you can't really just “add data” and it will still look icky.
I like that you said 250 ppi. It is very rare that you need more than 250. You don't need 300 in most cases. (Again, high quality artbook, etc.)