Usually the influence they have is in an indirect way. The lowering of the tax burden in the 80s paved the way for the boom in the 90s, the excesses of the 90s paved the way for today's downturn. Right or wrong most people give whoever's in office at the time credit (or blame) for what's happening.
I agree with what you said. Except presidents may initiate this legislation and nothing more: it becomes law only if it passes in Congress.
It is (i) Congress that influences the economy through expenditures and taxation (fiscal measures) and (ii) the Federal Reserve (monetary policy).
None of it belongs to presidents, whether Democratic of Republican.