I'm not sure how it was with Athlon 64 x2 but, at least now, max temp is max CPU temp, not core temp. The screen you linked shows core temps, so you likely aren't exceeding your CPU's specs.
I wouldn't agree with DM that stock cooler isn't enough if you aren't overclocking. It should be perfectly fine if your case has, at least, mediocre airflow and your ambient temperature isn't especially high. That is, if the cooler is more or less clean.