What does a 'full 18×16 barrel shifter' mean?
I also found it hard to find a concise description of this notation. However, by plugging in some different values for n and k in Google searches for "n x k barrel shifter", I was able to find this PDF, which shows an 8x4 barrel shifter on page 27, where there are 8 inputs and 4 outputs.

I'm not entirely sure what the value of an asymmetric barrel shifter would be, except perhaps just cost savings for a very specific implementation or device. They seem to be hard to find information about.

If you are in a class, I would ask your instructor or TA about what the correct interpretation is. The only resource I can find suggests it's inputs x outputs, but I think inputs x shift capability is a reasonable interpretation as well.