Are there common names for the subtiers of PTIME?
As complexity classes LIN, QUAD, CUB etc are not well-behaved, and not easy to handle.


In the definition, you must freeze the model of computation (that is, deterministic Turing machines with one tape). So you will do complexity theory in a single artificial model.
The definition of these classes is fragile: In contrast to our standard complexity classes, the classes LIN, QUAD, CUB do not remain stable if you move to other models, like RAMs or Turing machine with two tapes. 
There also is no natural reduction concept available on LIN, QUAD, CUB (except for reductions that are equally strong as the class itself). 


Summary: All in all this means trouble, and the classes are not very appealing.