I would say your exhaust water leak is a separate problem.
As others have said the DI problems are mostly associated with not enough voltage (usually REC/REG), or not enough fuel pressure. The fuel pressure problem is usually the fuel pump (or installation) and very rarely the air pump.
I would agree that your measurement gauge seems suspect, since the DI will not run at all below about 90 psi fuel pressure. It may run poorly between 90 and about 104, but it really needs to be 107 psi.
Regarding a low pressure fuel pump test: Sometimes the fuel pump Module installation into the fuel tank will allow the pump to "lean over" or "bend over" such that the "sliding seal" leaks, lowering the pressure. Make sure the fuel module is "square" before compressing the spring.
I have been running the High Flow Fuel 342DI pump in all four of my 2001 GTX DI for a few years now with good success. I agree the OEM pump is/was amazing (low current draw with great pressure), however, it does have some metal internals which do corrode with minor amounts of water in the fuel. Upon corrosion, the OEM pump either gets to low of pressure periodically (causing stalling or poor performance), or starts drawing so much power that the MPEM starves for power and therefore causes a variety of problems.
When I get one of the HFF 342 DI pumps that fails, it is often the pressure relief valve on the top of the pump. As a result I now always seal that with a bit of epoxy (JB Weld) before installation.
I am currently trying one of the HFF 500DI fuel pumps that is a bit noisier when priming but is almost the exact look and dimensions of the OEM pump. Therefore no "extension" is needed to make it the same height as OEM. This larger pump might be better and the noise of course can not be heard when running.
The OEM fuel modules and pumps were/are very expensive, and now you can only find used ones.