Different versions of Einstein's equivalence principle

As I understand it, there are two versions of Einstein’s equivalence principle. The first states that

“Locally, a frame in free-fall in a gravitational field is equivalent to an inertial frame in space in the absence of a gravitational field“.

The second states that

“Locally, a uniformly accelerating frame in space (in the absence of a gravitational field) is equivalent to a frame at rest in a uniform gravitational field“

I realise that these two statements should be equivalent (no pun intended), but it doesn’t seem immediately obvious to me and I’m hoping someone can explain?! (Is it simply that a frame in a gravitational field, accelerating purely due to the influence of gravity is equivalent to an inertial frame in free space, in the absence of gravity. This statement can be reversed, that is to say, a frame at rest in a gravitational field, purely due to the influence of gravity, is equivalent to a uniformly accelerating frame in free space?)

Furthermore, in the first case, I have seen elementary arguments where one considers a lift in free-fall, as follows: According to the observer on the ground, the net force acting on a particle within the lift is given by $mmathbf{g}+mathbf{F}=mmathbf{a}$ (where $mathbf{F}$ is the net force acting on the particle, other than gravity). The acceleration, $mathbf{a}’$ measured by an observer in the lift is related to the acceleration, $mathbf{a}$ measured by the observer on the ground by $mathbf{a}=mathbf{a}_{0}+mathbf{a}’$, where $mathbf{a}_{0}$ is the acceleration of the lift, and since it is in free-fall, $mathbf{a}_{0}=mathbf{g}$. Hence, $$mmathbf{g}+mathbf{F}=mmathbf{g}+mmathbf{a}’quadRightarrowquadmathbf{F}=mmathbf{a}’$$ and so the observer in the lift measures the same force acting on the particle as the observer on the ground, hence they are also in an inertial frame.

The problem I find with this argument is that the observer on the ground is not in an inertial frame themselves (since Earth is not an inertial frame). I’m I missing the point here though? Is it simply that this argument is used in the framework of Newtonian mechanics, where the Earth is considered an inertial frame, and so one can use this argument to motivate the equivalence principle (a believe this kind of argument was used by Einstein initially to convince himself of the equivalence principle)?!