I am seeing two definitions of strongly NPhard that seem to be slightly different:

A problem is strongly NPhard if a strongly NPcomplete problem has a polynomial time reduction to it.

A problem is strongly NPhard if it is still NPhard when all numbers in the input are bounded by a polynomial in the length of the input. Or equivalently, if it is still NPhard when the input is given in unary.
Wikipedia gives Definition 1, but Definition 2 seems to be more common overall. Both definitions preclude the existence of a pseudopolynomial time algorithm to such a problem. However, it seems to me that Definition 1 is stronger since there might exist a problem for which there is no pseudopolynomial time algorithm, but also no strongly NPcomplete problem can be reduced to it. This would be a strange problem indeed, but I don’t see why it can’t exist.
So am I missing something or is one of the definitions slightly off?