I tutor a cloud chamber workshop at CERN weekly, so I have some regular experience with recognising cloud chamber tracks.
There are only four particles (plus their antiparticles) we can observe in a cloud chamber:
- $\alpha$ (He nucleus),
- p (proton),
- $\mu$ (muon),
- $\beta$ (electron).
All other particles are either uncharged (and hence don't ionise the cloud) or decay too fast to survive at least a few millimeters (the minimal amount to be able to see them in a cloud chamber).
The width of a track depends on the ionising power of a particle, which in turn directly depends on its charge and its mass. From the above four particles, the $\alpha$ particle has charge +2 while the others have charge $\pm 1$. Their masses are roughly $m_\alpha = 4$GeV, $m_p=1$GeV, $m_\mu=0.1$GeV, and $m_e=0.0005$GeV. From this we estimate an $\alpha$ particle to be 8 times as ionising as a proton (with the effect of the double charge included), which in turn is 10 times as ionising as a muon, which finally is 200 times as ionising as an electron. This is of course merely a rule of thumb while the complete calculation is a bit more intricate. But the main result is correct:
The thick tracks are $\alpha$ particles or protons, while the thin tracks are muons or electrons.
But how do we distinguish between an $\alpha$ particle and a proton? Well, as you maybe remember from your high school nuclear physics class, $\alpha$ radiation can be stopped with a thin sheet of paper. In other words, there is no way that an external $\alpha$ particle could penetrate the casing of a cloud chamber. Hence, $\alpha$ particles that are observed in a cloud chamber are created inside. This can be due to an $\alpha$ source which has been positioned inside, but can also be due to natural background radiation: air contains a certain fraction of Radon, which is a natural $\alpha$ source. In our cloud chambers at CERN (which have no internal $\alpha$ source, so they only show the effect of Radon decay) this accounts to more or less 3 $\alpha$ particle tracks per minute (but the Radon concentration varies geographically, with the weather, and has some other factors, so it could be much more or less). Protons on the other hand cannot be created from radioactive decay, so they come from cosmic radiation.
Why does this matter? Because it is easy to distinguish radioactivity from cosmic rays based on their energy: while particles generated from radioactive decay have kinetic energies in the range of a few MeV's, cosmic rays have kinetic energies in the range of TeV - EeV (yup that's Exa-electronvolt, or $10^{18}$eV). This means that $\alpha$ particles, due to their huge mass relative to their kinetic energy, have very low penetration potential (that's why they are stopped by a sheet of paper) and can travel only a few centimetres in air. Protons are of much higher energy and can travel through the cloud unimpededly, and will hence form long straight thick tracks. Even though the length of a track also depends on the angle at which the particle crosses the cloud (implying that short thick tracks could be protons at a steep angle), it is known that very few protons are generated in a cosmic air shower (in one year of weekly tutoring workshops of 3 hours each, I have only seen 6 protons up to today). Furthermore, a last discriminating parameter is the particle's speed: cosmic particles move at speeds close to the speed of light, while (radioactive) $\alpha$ particles are so low in energy that they have speeds in the order of a few centimeters per second.
To conclude, the tracks from the video you refer to are thick, relatively short, and slow, so they are undoubtedly $\alpha$ particles.
Ps: the same reasoning can be applied to distinguish between muons and electrons. Muons can only be cosmic, while most observed electrons will be radioactive. Even though comparing the speeds won't be easy (even radioactive electrons move too fast for the eye to be called slow), we can make the difference based on the form of the track: long and straight implies high energy and hence a muon, curved and full of cusps implies lots of interaction so low energy, and hence an electron.