But the magnetic field in the relay's solenoid is in phase with the current in the "primary". Since the shorted turn is in that same field , won't the current in the shorted turn be in phase with the primary?
The magnetic layout is slightly more complicated than a "simple" transformer (with a single magnetic path linking 2 coils). The magnetic iron from the main coil is divided into 2 parts, one goes straight to the armature, the other has to pass through the short-circuited single-turn shading coil. Because of this, the shading coil has "leakage inductance" (formed by the magnetic flux path linking the shading coil, but not the main coil.) this inductance causes the current in the shading coil to lag the current in the main coil.
(If this lag didn't happen, the effect would be to just cancel a part of the flux linking the shading coil, causing a simple weakening or a "shading" of the flux, as described in your question.)
The shading current, being 180 degrees out of phase, retarded more by the leakage inductance, leads to the "imperfect cancellation". The flux in the shaded part of the core is weaker and delayed from the main flux. The cross-section of the shading coil has to "be right"; fat coil, too little resistance, lots of delay, but the delayed flux then would be weak. Thin coil, lots of delayed flux, but not much delay.
Think of an RC circuit excited by an AC source. If C and R are both large (so the time constant is much greater than the AC half-period) then nearly 90 deg delay, but almost no voltage on the cap. If C and R are small (time constant much less than AC half-period) then big voltage on C, almost in phase with applied voltage.
So in effect we have 2 separate relay coils and magnetic-path iron, each excited by AC of different phases, acting on a common armature, both trying to attract it with 120Hz "vibratory forces" (for a 60Hz relay). The armature "sees" the sum of these forces, which always add, never subtract (due to the square-law rule of magnetic attraction of iron to magnetic flux). The phase shift guarantees that when one is zero, the other is positive. So the holding force, while being still vibratory, never goes below some value. (In an ideal world, the phase shift would be 90 degrees, and the 2 fluxes would be equal in magnitude, giving a perfect vibration-free attraction). But such perfection is rarely achieved, and is not needed, since once the armature contacts the core face, only a small fraction of the pull-in current ("holding current") is needed to prevent the relay from releasing during the zero-crossings of the current in the main coil. This is why the cross-section of the shading coil can be a small fraction of that of the main coil.