Is specifically indicating "uncontrolled airfield, student handling radios" by writing "uncontrolled" in a student's log book necessary?
You need to specify that the requirements of training are met.  

For a private pilot under the FAA system, you need to have completed three solo landings at a controlled airport.  Otherwise, radio communication is a required line-item for training.  You should always log the airport of departure and arrival as well as the date.  If someone wants to know whether it was controlled or uncontrolled, they could look up the airport and see if it had a tower on that date, but they shouldn't need to since the required solo with an operating control tower should be specified in your training remarks.

Any examiner in the area would likely know whether any given airport is controlled or not simply from memory, being familiar with the area you're in. The examiner must check to make sure you meet the minimum requirements for the certificate or rating.  

Beyond the three landings with a tower and the required standard communications training, there is no need to specify controlled vs. uncontrolled airport operations.