The cooler which came with my Pentium D doesn't look like that and has the smaller copper spot. That doesn't mean much, given I got the chip ahead of launch and it wasn't boxed, so it might just be that "Arthur" included a regular cooler for single core chips.
Intel must have altered something though, as it proved adequate for the Pentium D at 2.80GHz. However, it was incapable of cooling a Core 2 Duo E8400 at stock clock to a satisfactory level, so if the Core 2 coolers were shorter they must have changed the materials or some other characteristic. I wouldn't know, because I have never owned a new Core 2 machine. My friend had one, but it died within minutes of him giving it to me and I have no idea what happened to the cooler, I suspect I either gave it to somebody or threw it away because I had no plans to do anything where I would need it, but I do remember it being shorter and I think it was an E6400 chip. The only Core 2 machines I have owned used non-stock cooling, one a cheap Akasa heatsink and another was on liquid cooling - the same liquid cooling the Pentium D was upgraded to and ran fine with, again showing a temp increase from the Core 2. Both ran hotter than hell and I always run the fans at 100% speed.
The reason I am writing this for anyone that finds this thread on Google later and uses the information, is to make clear that the size and shape of the cooler don't necessarily mean much. If you decided to use an Intel cooler with no idea what chip it was supposed to cool you are risking excessive temperatures. There will always be enough thermal mass that it is safe to do, but you should monitor the temperature under load for a while to make sure. The best solution is still, as was always known and likely already stated in here somewhere, to for out for a third party cooler - indeed, the Akasa proved to be just fine and wasn't annoyingly loud (At least, no louder than the Intel ones) and as the cheapest block of aluminum in the shop I can then only guess more costly coolers might perform even better.