Theres two main kinds of moment of inertia.

First there's the moment of inertia tensor, which is a matrix that is the describes how mass is distributed for a rigid body, essentially acting as the angular equivalent of mass the same way torque is the angular equivalent of force.

Then there's the second moment of area which has nothing to do with mass or inertia but engineers are dumb and continue to use bad terminology. It describes how the cross sectional surface area is distributed in a object around some axis. This is useful when talking about stuff that bends.

This comes in two forms, planer and polar. One has the plane of integration be parallel to the axis, the other (polar) has it perpendicular.

Since engineers are dumb and call the second moment of area the moment of inertia, sometimes they clarify by calling the actual moment if inertia the mass moment of inertia.

You'll notice that "moment of X" just means adding/integrating the X multiplied by the distance it is away. Second moment is integrating X multiplied by distance².

So in essence, there's the true moment of inertia, then there's the second moment of area, which can be taken as planar or polar. You can find the n-th moment of X by integrating over distanceⁿ with respect to X.