Think of the budget as a multi-year allocation. If Budget A from 2 years ago was "$500 this year, $600 next year, $700 the year after that" and now you reduce the allocation to $610, that is a budget cut from the original Budget A.
But this is more about having an expectation of funding at a certain level from whatever source, and knowing that that will be your "budget" (now meaning a spending _limit_), then having that amount come down. You now have a smaller limit.