Hamilton's Principle is that the first order variations of $\displaystyle\int^{t_2}_{t_1} L$ $ dt$ for an on-shell trajectory in the configuration space should vanish provided the varied off-shell trajectories meet the on-shell trajectory at $t_1$ and $t_2$.
The only reason I can think of as to why this should not be called the Principle of Least Action is that it is not - it is the Principle of Stationary Action. But leave that distinction aside for a moment because it is sort of irrelevant for the doubt that I have, as will become clear in the rest of my post.
The issue that I have is that my Professor is insisting that the condition obtained on the ''$\Delta$ - variation" of $\displaystyle\int^{t_2}_{t_1} L$ $ dt$ about an on-shell trajectory in the configuration space for a system with conserved Hamiltonian (both on-shell and off-shell) is exclusively what should be called the Principle of Least Action. This condition is that $$\Delta \displaystyle\int^{t_1}_{t_2}p\dot q dt=0$$ where the $\Delta$-variation is defined as the following: $$\Delta \displaystyle\int^{t_1}_{t_2}p\dot q dt=\displaystyle\int^{t_1+\Delta t_1}_{t_2 + \Delta t_2}p\dot q dt + \delta \displaystyle\int^{t_1}_{t_2}p\dot q dt$$ with a key difference from the variations considered in the Hamilton's principle that apart from varying the limits of integration as well, we do not require the varied off-shell trajectories to coincide with the on-shell trajectory at the end-points of the on-shell trajectory.
Now, I have always read that $\displaystyle\int L dt$ is what is called action and thus, I think that a variational principle associated with the same should be called the Principle of Least (Stationary) Action. Whereas $\displaystyle\int p\dot q dt$ defer from $\displaystyle\int L dt$ by $\displaystyle\int H dt$ - jeopardizing the possibility of its variational principle's to be called "Principle of Least (Stationary) Action" (leave alone an exclusive right to be called so).