I am seeing two definitions of strongly NP-hard that seem to be slightly different:
A problem is strongly NP-hard if a strongly NP-complete problem has a polynomial time reduction to it.
A problem is strongly NP-hard if it is still NP-hard when all numbers in the input are bounded by a polynomial in the length of the input. Or equivalently, if it is still NP-hard 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 pseudo-polynomial 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 pseudo-polynomial time algorithm, but also no strongly NP-complete 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?