How to classify devices into photodiodes, photodetectors and photocells?
There are a lot of types and each has benefits and drawbacks.  The wikipedia pages have a lot of informative reading available.

Photodiodes would 'generally' be considered faster switching than photocells but photocell is not a very accurately defined term referring mostly to 'solar cells' that are intended for more output but not fast response.

https://en.wikipedia.org/wiki/Photodetector