For the elliptical wing, which property is actually elliptically distributed?
It is both the planform and the circulation distribution. Note that circulation is not lift coefficient but bound vortex intensity. You can interpret it as local lift coefficient times local chord.

On the untwisted elliptical wing the local lift coefficient is constant over span, and changes in angle of attack over the linear range will change the lift coefficient equally everywhere. This, when combined with the elliptical chord distribution over span, means that the circulation distribution will stay elliptical over the linear angle of attack range. This is the special characteristic of an elliptical wing: While any wing can have an elliptical circulation distribution at one angle of attack (given the right twist distribution), the elliptical wing will keep that elliptical circulation distribution over the whole operating range.

With an elliptic circulation distribution comes also a constant induced angle of attack and downwash angle over span. I guess this is expressed by some authors with the term "regular".

However, only the aerodynamicists will see that as an advantage. Both weight and stall characteristics of elliptical wings are less than optimum; the low induced drag coefficient is bought with higher structural mass and, consequently, lift. A more triangular circulation distribution will yield the lowest wing weight and overall drag for a given non-lifting mass (that is, all mass that is not involved in lift generation, especially the payload). Note that for such a triangular distribution drag will be highest near the center.

When people talk about elliptical lift distribution, they mean lift per span. I prefer to use the more correct term circulation, since lift is a force as in pressure times area and can only be produced by a whole wing or at least wing section, not one spanwise station.