Does not sound like the physical fuel pump is bad.
The "noise" is the fuel pump. When you turn the key to "on" the fuel pump primes. This takes a second or two.
If you "start" the car the fuel pump still has to prime before the car will start. It will run on lower fuel pressure, just not well.
IIRC there is a TSB for long crank/hard starts that Ford can do. It may help some. It's an ECU reflash and will cost money. How much I can't say. Might be enough that it may be worth looking into an Xcal and custom tune.
But just to cover bases,
When was the Fuel Filter last replaced?
If you don't know or it's been a while, replace it. It's cheap and relatively easy to do. It also rules out 1 possibility.
Do that first and see if anything changes or improves.
13 years, 124,000 members and still no sponsored racing.