Amortized analysis - adding operations to a data structure
I don't know much about the amortized running time of binomial heap, but after reading Wikipedia, I suspect the following might be true: the amortized time of a long sequence of consecutive insert operations is $O(1)$; but this is only true for consecutive insert operations that don't have any other operations mixed in.  Based on that, I wouldn't really summarize the running time of a binomial heap by saying that insert takes $O(1)$ amortized time, without further caveats.
As you quite astutely point out with your example based on a binary counter, the amortized running time of an operation depends on the full set of operations that are allowed to be performed.  So I suspect that the resolution is that the claim about the amortized running time of inserts into a binomial heap wasn't quite right, or had some additional caveats.  I don't know that for sure, as I don't know much about binomial heaps, but that would be my running hypothesis at this stage.