As far as I know, moment of inertia is the distance between the point the force is applied and the point where the object rotates around. In that case, it wouldn't be any different between the two.

The [perpendicular] distance between the point where a force is applied and the axis of rotation is the "moment arm". If you multiply the magnitude of the force by the length of the moment arm, the result is "torque".

The moment of inertia is a measure of how an object will accelerate under an applied torque. The greater the moment of inertia, the more torque (or the more time) it will take go achieve a particular rotation rate. It can be computed by considering an object as a bunch of tiny pieces, taking the mass of each piece multiplied by the square of its distance from the axis of rotation and adding up all of those results.

An object with a large mass or with a mass that is very spread out will have a high moment of inertia. It will take a lot of torque to change its rotation speed.

The perpendicular distance between the point the force is applied and the point the objects rotates around (center of mass) given the centers of mass of the objects are in the same place

The original question is not well enough posed to have a definite answer. But yes, for a reasonable understanding of the intended question, the answer is "the middle".

When you give an impulse to one end of the rod, the COM moves forward and the object rotates around the COM. The initial velocity of the other end if the rod is the sum of the forward velocity of the COM and a backwards velocity from the rotation.

Question: what mass distribution will result in the other end of the rod initially moving backwards from its starting position with the greatest speed?

Answer: undefined. There is no optimum. The distribution with all of the mass exactly in the center results in an undefined speed and, accordingly, is ineligible.

Do you mean that the location of center of mass does not matter for this, the only thing that does is how "centralized" the mass is to CoM, irrespective of it's location. In short: Make every where that is not CoM make as light as possible.

The location of the center of mass relative to the ends matters. But so does the degree to which the mass is centralized. Both are relevant. One can even write an equation.

Try it. Apply an impulse "p" at right angles to one end of a rod of length "l" and mass "m" with center of mass offset "r" from the end where the impulse will be applied. If the rod has a moment of inertia "I", what speed will the other end have as a result?

Edit: Take it a step at a time. For instance, what rotation rate will result from the applied impulse?

If the location of the center of mass relative to the ends matters, would the optimum position be in the middle, or closer to the side which the force is applied?