In general, these puzzles are called "Deletion Puzzles" but the usually do not include the restriction to the beginning or end letters. The most famous example is "startling" as it is the longest. It is also sometimes called "Elision". Please not that these terms also do not require it to be iterative.
"Terminal Deletion" or "Terminal Elision" refer to removing both the first and last letters to make a work.
"Decapitation" word puzzles refer to only removing the first letter to reveal a word.
You would most likely have luck calling a puzzle regarding these words an "Iterated Amputation" word puzzle. Amputation usually has the same definition as Decapitation for this context but it less common. It would be easy to draw contrast between them and describe removing either of the two "arms" of the word (right or left arm). Iterated, of course, refers to the fact you want to do this several times. Does that mean you call a word "amputatable"? I don't know.
Ultimately I don't think a single term for what you want exists and, if it did, I can't think of a context where you wouldn't have to define it for the audience anyways.
I suggest you check out "A Key to Puzzledom: Or, Complete Handbook of the Enigmatic Art" which is available on google books.