I refined my post in response to having it redirect to another question which addressed the original form of the question.
In Classical mechanics with calculus of variations, and optimal control : an intuitive introduction by Mark Levy, the energy of a conservative one DOF system is given in (1.30) as
$$H\left(x,p\right)=\frac{p^2}{2m}+U\left(x\right).$$
We are then told:
The energy thus expressed in terms of position and momentum is called the Hamiltonian of the system.
The sentence seems to refer specifically to (1.30). Nonetheless, it is well-known that the Hamiltonian can be given in more general forms. So it seems that his assertion could be understood to mean: Any Hamiltonian of the system is equivalent to the energy given as a function of momentum and position.
Is that a correct statement?
I am intentionally being vague regarding the definitions of momentum and position.
