Welcome to the forum and to our humble little slice of it!
While c500 fan is right to question the type/brand, as there CAN be compatibility issues, I think it's more likely that you're running 445605-001. The sticker you have posted the picture of is simliar to stickers on my boards, the part number isn't on that one. I'm not sure where your label might be, but you can check in the hard drive bay too as I have seem them stuck there as well on other systems.
To the matter at hand... That it won't even POST with 4 GB installed suggests 445605-001 and the i943GML. If it were 441696-001 (i940GML) it should POST with 4 GB but only use 2, and of course 441695-001 (i945GM) should POST and use 4 GB.
This is just my experience with the systems, but I have confirmed the behavior of these chipsets in other similar systems. For example: I can conclude safely that the iMac I have is an i943GML as it will run a Core 2 Duo but will absolutely NOT post if more than 2 GB of RAM is installed. Interestingly, Apple's own documentation about system capabilities is more clear and open about this behavior than even Intel's.
c500 fan's idea to test one stick is good, as the system should post with just one 2 GB installed regardless of chipset. That would at least tell you that both sticks are GOOD and that they'll work in the system. After that, if they both POST separately, it's a sure bet you have 445605-001.
For the record: Do NOT trust the information on HP's parts website except as a VERY loose guide. The information is routinely inaccurate, sometimes hilariously so.They list all three variants of motherboard for these systems as running 945 chipsets, which is absolutely not true as a magnifying glass, strong light source, and your own eyes can tell you from looking at the chip directly.Or a few minutes thumbing through this and other forums (though of course this one is the best!). Parts sites and places like eBay are also suspect as many of them just pull their descriptions from HPs parst info which, as I said, is often inaccurate.
Let us know how your tests come out, I always welcome the opportunity to be wrong!
O Shadow Fox O
EDIT: For grins, I pulled out my 445605-001 board and checked the label that looks like yours above NUMCRUN. While most of the information doesn't match 100%, the third barcode down has the same number up to the LO. Mine ends in LO5. It may not mean anything, but I thought it was worth at least mentioning.