Red giants and asymptotic giants have some close similarities, and one actually evolves into the other. Both have an extended envelope of relatively cool, non-burning material (mostly $\rm{H}$, $\rm{He}$). They also each have a core of dense, non-burning material; in the case of the red giant this is mostly $\rm{He}$, while for the asymptotic giant it's $\rm{C}$ and $\rm{O}$.
The burning shell in the red giant is $\rm{H}$. For stars of the right mass, the conditions (density, temperature) in the core will periodically be sufficient to ignite the $\rm{He}$ causing a "core flash".
Red giant structure:
After the red giant branch of stellar evolution there is a brief period where the $\rm{He}$ core burns called the horizontal branch. Once the He core is exhausted (it's been converted to $\rm{C}$ and $\rm{O}$), the star starts on the asymptotic giant branch. This branch has two parts, the early asymptotic giant branch (E-AGB) and the thermal-pulse asymptotic giant branch (TP-AGB).
E-AGB structure:
Stars on the E-AGB are like red giants, but in addition to a $\rm{H}$ burning shell there is a $\rm{He}$ burning shell (the energy output is dominated by the He burning shell). In the TP-AGB, the $\rm{H}$ shell picks up again and dominates the energy output, but periodically as the $\rm{He}$ produced by $\rm{H}$ burning is accreted onto the $\rm{He}$ shell, "helium shell flashes" occur, analogously to helium core flashes in red giants.
Source/Reference: Carroll & Ostlie "An Introduction to Modern Astrophysics: 2nd Edition" (Pearson)
Best Answer
The short answer is no: there isn't a agreed theoretical distinction between red giants and red supergiants.
The source of confusion is that the red giant branch (RGB) is a theoretical concept, but the Yerkes spectral classification also defines a red giant observationally. So when astronomers talk about red giants, it isn't necessarily clear which is meant. Scientists who calculate models of stars are probably talking about stars that are burning hydrogen in a shell around a helium core, whereas scientists who actually measure starlight are probably talking about stars which meet certain criteria for colour and brightness. The spectral classification defines red supergiants, but there isn't a corresponding theoretical "red supergiant branch" like there is for the RGB.
So, unlike red giants, red supergiants are simply bright, red stars. It so happens that they may be in the same evolutionary state, but it is also possible that they have moved on. For example, most massive stars will appear as red supergiants while helium is fused into carbon in the core.