Brief overview of CID vs. HCD
"Collision-induced dissociation" or CID is a much older and more general term in mass spectrometry than HCD. HCD is a vendor-specific term invented to describe a new modification on how ions can be dissociated in Orbitrap mass spectrometers. CID is a "universal" term that applies to many different types of mass spectrometers. Unfortunately, how exactly CID works depends on which type of mass spectrometer you are talking about.
Wikipedia has a good overview of how CID implementations vary according to instrument type. Understanding that the meaning of "CID" varies strongly by instrument type is important. The naive assumption that CID on e.g.a triple-quadrupole or qTOF instrument is like CID on an ion trap instrument is understandable -- it is the same term after all -- but very wrong.
CID on an Orbitrap
On an ion-trap instrument like your Fusion instrument, CID is done in an ion trap. This means that on an Orbitrap, CID is a resonant excitation technique. Thus, to reach a certain collision energy, ions must be accelerated relatively slowly from their starting energy to the final energy. (The final energy is a user-controllable parameter.) They thus collide many times with the inert gas during the excitation process. Each of these collisions may "heat" or "cool" the ion. In an ion trap, resonant excitation will only accelerate the precursor ions, so after dissociation has taken place, fragment collisions with the gas molecules will cool the ions, eliminating or at least reducing further fragmentation into smaller pieces.
A 2014 paper in the Journal of Mass Spectrometry by Ichou et al. explains CID on an ion-trap-equipped Orbitrap fairly well:
In ion trap devices, each collision can activate or cool the stored precursor ion. The evolution of this equilibrium depends greatly on the ion kinetic energy. The second step relies on the unimolecular dissociation pathways of the precursor ion. It has been extensively modeled using the Rice–Rampserger–Kassel–Marcus (RRKM) theory18-21 and the quasi‐equilibrium theory.22 Note that the internal energy distribution of ions greatly relies upon the rate of these two steps. So, the internal energy distribution depends on the regime of collisions under study.23-25 The ‘slow’ or ‘rapid’ heating regime is defined according to the unimolecular dissociation constant rate. If the latter is slower than the constant rate of the activation/deactivation process, the regime is defined as a slow heating technique. The slow heating regime in an ion trap instrument for the conventional CID mode may be described as a random walk process,25, 26 where a large number of ion‐neutral gas collisions are needed to ‘heat’ the ions in order to dissociate them. Some studies25-29 have demonstrated that the ion internal energy distribution for ion trap instruments is close to the temperature due to the multiple collision regime.
Thus, on an ion-trap Orbitrap like your Fusion, CID can be at least close to an equilibrium process. Whether it is or not depends on how quickly ions are ramped in energy to the final user-decided collision energy. If ramp times are long compared to the unimolecular dissociation rates, ions have the opportunity to distribute the excitation energy across most of the bonds in the molecule. They bump into many gas molecules, sometimes being heated by the collision, and sometimes being cooled. These collisions are what allows the energy to distribute itself among all the bonds of the excited ion. As a result, in slow resonant CID, the weakest bonds break first. If ramp times are short, then CID (on an ion-trap instrument) ions do not have time to equilibrate and dissociation spectra can be more like what is observed for HCD.
HCD on an Orbitrap
The same paper explains HCD fairly well too:
'HCD' is an excitation mode that belongs to the category of non‐resonant activation techniques like the CID conducted in tandem‐in‐space instruments.
Thus, 'HCD' in an Orbitrap is like CID on a triple-quadrupole or qTOF instrument. It is a non-resonant dissociation technique. Ions do not experience a long series of "equilibrating" collisions with gas molecules. Effectively, a single bolus of energy is imparted to them very quickly. This type of dissociation is a non-equilibrium process. Both weak bonds and stronger bonds may have a chance to break, depending on which ones are excited by the smaller number of collisions with gas molecules. Product ions remain excited, and do not experience enough collisions to "cool" them down, meaning that if they do further collide with gas molecules, they may be further fragmented.
Answers to your questions
Is HCD design to keep the fragmentation products in the trap so that they collide many times with the inert gas?
No, actually the opposite, as hopefully the discussion above makes clear. On an ion-trap instrument (but not on other instruments), it is CID that collides ions many times with the inert gas.
When I apply HCD on inorganic molecules I observe more fragmentation products than with just CID suggesting multiple fragmentation steps. However I don't get why physically this is possible.
The reason HCD gives you more fragment ions is that ions do not have time to equilibrate, so many bonds can be broken, each with some probability. Parent ions are energized rapidly, in a near-single-step, to highly excited states. A variety of vibrational modes can get excited, leading to a variety of bonds breaking in the dissociation step. In (slow) CID, ions are excited slowly; as energy accumulates during the resonant excitation, ions collide with and have time to redistribute energy over the whole molecule, meaning that only the weakest bonds tend to break, resulting in fewer fragments.