Based on online specifications your system has the Intel® C206 Chipset, which requires Intel® Rapid Storage Technology Enterprise.
It is strongly recommended to use the driver provided by HP*.
Yes, the Device Manager shows a couple of things:
under Storage Controllers there is one item: "Intel(R) Desktop/Workstation/Server Express Chipset SATA RAID controller"
and under System Devices there is an item: "Intel(R) C206 Chipset Family LPC Interface Controller - 1C56"
In fact, at the link you have kindly provided there are three versions of IRST under Driver-Storage. I tried them all, and they all fail in the same way. The "Problem Signature" I provided above is for version 10.6.0.1002. Versions 10.5.0.1027 and 10.1.0.1008 produce a similar signature. So, I'm still stuck.
You may try these installers; otherwise, contact HP since the drivers they provide should work with your system.