Exactly!
And as you can see, the language recognizes the position of the Treasury within the executive branch so that it may constrain the president's hand, requiring him to have permission to spend.
If it was Congress doing the spending then that phrase wouldn't make any sense. There would be no consequence of appropriation requirement, it would simply be saying that Congress spends.
So yes, the Constitution is clear as it defines the interaction between the executive and legislative branch, recognizing that the president must have permission to spend, as he spends.