Said by other words, think of what the equation $a=R\alpha$ represents: $a$ is the acceleration of a particle on the rim (a distance $R$) from the centre).
On a simple freely spinning wheel, the rim is moving with $a$. On a wheel of a car, the rim is actually standing still at the point of contact with the road. Here instead the centre is moving with $a$. And the opposite rim-point moves with $2a$.
The point is that this $a$ is the motion (acceleration) of a rim-point relative to the centre.
Back to your situation. Your strings are hanging still. No acceleration after unwinding. So the final touching point where the strings touch the cylinder is standing still in that moment, just like the contact-point of wheel-on-road for a car. This is the same situation as the car's wheel, where it is the centre instead of this rim-point that accelerates with $a$. And the opposite rim-point has $2a$.
The rim-point moves with $a$ seen from the centre-point and because of that the relation $a=R\alpha$ holds.