Earlier quoted context omitted.
> President writes the budget The President proposed budgets. The House and the Congress have the power of the purse.
The president is part of the legislative branch because they have veto power. It's really not as simple as "the president writes the budget" or "congress writes the budget". The entire legislative branch writes it together.
Veto is a check on the legislature by the President. It does not make the President part of the legislature anymore than it makes every district court judge. (The Vice President has a hybrid role.)
Congress can pass a budget with zero executive input by overriding the veto. One can correctly say the President is part of the legislative process. But not the branch.