reference frames - Is the lay explanation of the equivalence principle wrong?

The common explanation/trope for the equivalence principle always has something to do with you being inside an elevator or spaceship, and your supposed inability to differentiate, say, gravity from a rocket, or free-fall from being motionless in empty space. For example, Wikipedia says "being at rest on the surface of the Earth is equivalent to being inside a spaceship (far from any sources of gravity) that is being accelerated by its engines." Many other sources say similar things, often making broad, sweeping generalizations like "there is no measurement you can make to distinguish the two scenarios."

Isn't that just plainly wrong? Gravitational fields aren't homogeneous. Here on the Earth, a clock on the floor runs more slowly than a clock on the table, and we have clocks precise enough to measure such small differences due to the gravitational gradient. But doesn't a clock in an accelerating spaceship run at the same rate no matter where in the ship you put it?

I searched the top 50 or so questions about the equivalence principle and found this answer that talks about tidal effects in gravitational fields, but the explanation is very confusing. As far as I can tell, it seems to be saying the "attraction" between the particles that arises as a result of slightly different gravity vectors is somehow equivalent to the actual mutual gravitational attraction the particles should have absent the external gravitation field. I don't see how that could possibly be the case.

Also in that answer is a link to this explanation of tidal forces and the equivalence principle, which seems to be saying the opposite: that tidal forces are, in fact, a distinguishing characteristic between gravitational acceleration and rocket acceleration, and that the equivalence principle only exactly applies to small enough points in space over a small enough duration, where tidal effects are negligible.

I know what seems most correct to me, but as I'm not an expert in this field, would an actual expert please shed some light on these varying views of the equivalence principle?

Answer

But doesn't a clock in an accelerating spaceship run at the same rate no matter where in the ship you put it?

Remarkably, the answer is, even in the context of SR, no.

It turns out that acceleration of an extended object is quite subtle.

That is to say, we can't meaningfully speak of the acceleration of an extended object.

Essentially, the 'front' (top?) of the spacecraft accelerates less than the 'back' (bottom?) of the spacecraft if the spacecraft is not to stretch and eventually fail structurally.

Thus, the clocks at the front (top) run faster than the clocks at the back (bottom) as would be the case for clocks at rest at different heights in a gravitational potential.

This is actually well known and best understood in the context of Rindler observers.

Note that Rindler observers with smaller constant x coordinate are accelerating harder to keep up! This may seem surprising because in Newtonian physics, observers who maintain constant relative distance must share the same acceleration. But in relativistic physics, we see that the trailing endpoint of a rod which is accelerated by some external force (parallel to its symmetry axis) must accelerate a bit harder than the leading endpoint, or else it must ultimately break.

Now, this isn't meant to answer your general question but, rather, to address the particular question quoted at the top.