This is two different things, not helped by the multiple meanings of the word budget. If the organization makes the budget (i.e. a spending plan) without confirming the funds (or knowing them through, say, a long term understanding), and don't get what they seek, of course that's not somebody else "cutting" their budget. That's just a budget you made up not being agreed to.

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.