Equilibrium position

A place, not a state. Equilibrium names the condition of zero net force; the equilibrium position names where that condition holds.

EK 7.1.A.2.ii is one sentence long: an equilibrium position is a location at which the net force exerted on an object or system is zero. Everything else about oscillations is measured from there. EK 7.3.A.1 writes SHM as the displacement measured from its equilibrium position, $x = A\cos(2\pi f t)$ or $x = A\sin(2\pi f t)$, and amplitude is the maximum value that displacement reaches.

It is not always the spring's relaxed length. Hang a block on a vertical spring and the net force is zero where the spring force balances the weight, below the relaxed end. Measure $x$ from the relaxed length instead and every term in the energy and force analysis is offset. EK 2.8.A.3 points the same way: the force exerted on an object by a spring is always directed toward the equilibrium position of the object-spring system.

Passing through it is the fastest moment, not the slowest. Zero net force means zero acceleration, which is not zero velocity. The restoring force has been pushing the object toward this point the whole way in, so the speed peaks exactly here and the potential energy bottoms out. Reading "equilibrium" as "at rest" inverts the whole graph.

It is also the reference for the restoring force. A restoring force is defined as opposing displacement from this position, so naming the wrong point makes the force look like it is pushing the wrong way.

AP Physics C: Mechanics uses the same anchor for resonance: EK 7.3.A.4.iii calls the natural frequency the frequency at which a system will oscillate when it is displaced from its equilibrium position.