I had some trouble finding this information, so I believe it would be interesting to have a post in here explicitly mentioning the issue described on the title. Furthermore, I would like to ensure my line of thought is correct.
In Classical Mechanics, it is simple to show that the Euler-Lagrange Equations for a Lagrangian $L$ dependent on second derivatives of coordinate - i.e., $L = L(q, \dot{q}, \ddot{q}, t)$ - are of fourth order. The Einstein-Hilbert Action, when expressed (incompletely) in the form $$S = \int R \sqrt{-g} \text{d}^4x$$ have a Lagrangian density dependent on second derivatives, and hence one would expect the EFE to be of fourth order. Nevertheless, they are of second order.
In one of Domenico Giulini's lectures on the W-E Heraeus International Winter School on Gravity and Light, I recall he mentions that the E-H action must be supplemented by a surface term, for otherwise the expression is not functionally differentiable. Furthermore, in Eric Poisson's "An Advanced Course in General Relativity", it is mentioned (right after Eq. (4.1.18)) that the presence of second derivatives on the E-H Action (in the form stated above) requires the presence of a boundary term.
My question is then the following: are the EFE of second-order because the "true" E-H Action includes a surface term precisely chosen so as to cancel out the second derivatives contributions to the EoM? Furthermore, I believe most usual courses on GR won't mention this boundary term, but get to the second-order EFE either way. Why does that happen? Is it a cleverly hidden miscalculation or are these different contexts?
I believe there is a chance the answers might already be contained in this question, but I don't want to provide an answer to my own question because I am not sure if the reasoning is correct.
