This is almost right. Since Mars is so much smaller than the Earth, its iron core has cooled and hardened. That killed the volcanoes but more importantly, it killed the magnetic field, which is what protected Mars from most of the solar wind streaming constantly from the Sun. That solar wind slowly blew away the Mars atmosphere into space.
The Sun's heat varies by less than .5 degrees over the last million years. But the Sun's magnetic field, which stretches to all the inner planets, is quite variable. When the Sun is more active, during the height of the sunspot cycle, it's magnetic field shields the Earth from slightly more cosmic rays.
On the quiet side of the Sun's magnetic cycle--no sunspots = less magnetic field--more cosmic rays get to the Earth's atmosphere, which are powerful seeds for cloud formation. More average cloudcover acts to cool the Earth, even to the point of a mini ice age like we had 200-300 years ago when sunspot activity virtually ceased.
As we've said, this climatology stuff is quite complex; much deeper than the simplistic "global warming" hypothesis that beguiled me and everyone else thirty years ago.
Mars has a much thinner atmosphere because it is smaller and has less gravity as well. I wonder if terraforming Mars would be feasible given the lack of a magnetic field to deflect the solar wind.