Earlier quoted context omitted.
The executive requests the budget. Congress approves it. The key thing here is the "Due Care Clause." The administration is required to implement our laws with "due care." If there's a law on the books the administration is required to enforce it. The means and expected effectiveness of that enforcement are often not specified.
The House creates the budget, the Senate votes on it, and the president then signs it into law. Usually, the executive branch will come up with a dream budget that can be considered their starting position in budget negotiations.
As President, Trump paid no price for shutting down the government because the budget was too small/didn’t include some inefficient spending he said Mexico would pay for. His next will be his fourth shutdown.