When exactly does a sub giant become a red giant? A simple, but interesting question I think.
I understand that after the hydrogen exhaustion on the main sequence the post main sequence starts with the SGB where they burn hydrogen in a shell around the Helium core.
We also know that stars under $ 2 M_{\odot }$ experience a Helium flash and start Helium core fusion after the Red giant tip. Whereas stars greater than $2 M_{\odot }$ do not experience this Helium flash and starting the He-core burning immediately after the SGB without a helium flash (very massive stars burn Helium on the main sequence).

My question is now what specific physical process happens in a sub giant, which makes it a red giant afterwards?

Thank you in advance (I am very open for mathematical/physics explanations as well, since I study physics myself).
The straightforward definition is in terms of where a star lies on its evolutionary track in the HR diagram (see below). The subgiant branch stars are those which have exhausted their hydrogen core and which are burning hydrogen in a shell but their He cores have not begun to contract significantly. The distinct upturn in luminosity marks the beginning of the red giant branch. This occurs when the core grows significantly more massive, cannot support itself hydrostatically and begins to contract. At the same time the envelope expands and becomes convective and the H-burning shell moves inwards and increases in temperature and luminosity.
The tip of the red giant branch is where He is ignited. This takes place &quot;explosively&quot; in a degenerate core if the star is below about $2M_{\odot}$ (not $5M_{\odot}$), but begins smoothly in higher mass stars. This causes the core to expand, pushes the H-burning shell out and reduces the luminosity.

HR diagram showing the location of the subgiant and red giant branches on the evolutionary tracks for stars of different mass. The tracks begin at the main sequence and evolve to the termination of the main sequence, across the subgiant branch, turn up the red giant branch, which terminates when core He burning commences and the stars become less luminous and hotter.