The change appears to be clarifying that if the 15 NM distance determination is based on ground distance, for example the aircraft has passed a fix which is known to be farther than 15 NM from the VOR, that is acceptable. The increased minimum distance only applies when the distance is measured using DME.
This is because DME measures slant-range distance between the antenna in the aircraft and the transponder on the ground. The calculated distance is the hypotenuse of a right triangle whose other sides are the actual over-the-ground distance and the height of the aircraft above the DME transponder. This hypotenuse will by necessity be greater than the over-the-ground distance. For example, when the aircraft is 15 NM from the DME transponder and at FL190, the DME measurement will be:
$$
\begin{align}
\mathrm{DME}^2 &= x^2 + h^2 \\
&= \left(15 \mathrm{~NM} \cdot \frac{6076 \mathrm{~ft}}{1 \mathrm{~NM}} \right)^2 + \left(19000 \mathrm{~ft} \right)^2 \\
\mathrm{DME} &= \sqrt{ 91140^2 \mathrm{~ft}^2 + 19000^2 \mathrm{~ft}^2 } \\
&= 93100 \mathrm{~ft} \\
&= 15.3 \mathrm{~NM}
\end{align}
$$
Whereas if the aircraft is at FL450 above the exact same spot, the DME measurement will be 16.7 NM; at FL600 the measurement will be 18.0 NM. You can see that if the intention is to ensure that an aircraft is at least 15 NM from the VOR over the ground, and if DME is used to measure that distance, you need the DME measurement to read greater than 15 NM—and the amount by which it must be greater increases as the aircraft's altitude increases.
You will also note that even at very high altitudes the required DME distance is lower than what is prescribed in Doc 4444. I would guess that ICAO wanted to add some buffer to the allowable minimum distance in the interest of safety.
