Not picking on you, just this statement comes up more than the correct statement does.
The house has dibs on origination of TAXING (bills for raising revenue), not on spending. Obviously the House has to agree to the spending.
Your second sentence above must be added to the power to acquire revenue (raise revenue). Seldom will there be the notion "let's just raise a billion dollars" without some reason attached for doing so, even if it's simply "let's raise a billion to have sitting in an account." The purpose is: "to have sitting in an account" in that instance.
And, the power to block any other use of that money is, as you say, contained in the requirement that the Senate and House must agree on any bill going forward.
So far as my comment that "disbursing funds" is the domain of the House, I would have to agree with you in terms of the exact wording of the Constitution. However, the "purpose" for raising those funds is generally assumed in the raising of them.
The bottom line is that for the most part I agree with you. Thanks for your thoughts. I appreciate precision, and you forcing me to be precise. That's a good thing.