Okay good and thanks.
The fuel pump: I'm asking about the lift pump, and not the injection pump. I'm not trying to insult you here, but I don't really know what you do and don't know, so bear with me.
The injection pump is the one with steel lines that go to the injectors on the engine. Is this the pump that looks new? If the machine will run for ten minutes, this pump is probably okay.
The pump I'm concerned with is the lift pump. It may be electric or mechanical and bolted to the engine block, but it is in between the fuel lines from the tank to the injection pump.
If you pull the line from this pump, leading to the injection pump, if it's working, you should get a stream of fuel flowing out: if it's mechanical, you have to crank the engine, if it's electric, you turn the key on. It should be a good steady stream.
So, we need to know if the lift pump is mechanical or electric, and whether fuel is flowing when you do this test.
If it's a mechanical pump, and not flowing well, you probably need a new lift pump, but you could have a leak or restriction in the line going to your fuel tank.
If it's an electric pump, and you don't have a good flow, it may be a bad pump. If there is NO flow, it may be a bad pump OR an electrical problem with no power to the pump.
I'm still not sure you don't have an electrical problem, seeing as you stated that the machine shuts down like you keyed it off. Fuel problems tend to make the engine die after dropping rpm.
You also need to check the fuel line from your tank to your lift pump to the enjection pump, and the return line from the engine back to tank for any leaks or kinks.