The issue is solved and I will post my findings on here for others that need it in the future as this might well become the ultimate 945 series chipset iAMT issue thread on the internet.
Out of the 6x brand new factory sealed D945GTPLKR (AAC97837-306) desktop boards with stock 4076 BIOS with build date April 2007 that I powered up with the same hardware and OS, I got the following results:
1x board exhibited the above conditions no matter which BIOS I used or no matter how many how many versions of drivers I tried and I will consider it as a defective board.
2x boards exhibited the above conditions with the stock 4076 BIOS, but the condition went away after I flashed it to the latest 4131 BIOS.
3x boards loaded correctly with the stock 4076 BIOS remained the same after I flashed it to the newest 4131 BIOS.
The chipset and LAN drivers that Gize linked works perfectly provided the board is not defective. The drivers provided by Intel on their support page for the D945GTP series board is not recommended as they are newer and more troublesome to install.
Chipset (INF_9.4.0.1027.exe): https://downloadcenter.intel.com/Detail_Desc.aspx?DwnldID=23215
LAN / AMT (intel_msft_v14.2_whql_rev1.zip): https://downloadcenter.intel.com/Detail_Desc.aspx?agr=Y&ProdId=1062&DwnldID=18236
After this experience, I really don't know what to think, how can there be this many variations of the same issue to the same desktop board with the same hardware and OS ?
I still have an additional 6x D945GTP boards waiting to be turned into systems and I will try to post additional results once I get around to it.