I'll have to take a closer look at the current version of the article, because I have an uneasy feeling that the answer actually is found in one of the subsidiary articles, such as Lydian mode. The short answer is that the connection between the Greek harmoniai and the medieval modes is more tenuous than you appear to believe, and the connection between medieval and modern modes is fairly straightforward. The "modern system" is actually that of Common Practice Tonality, and "modal" usage within that context begins to establish itself as a separate idea during the 18th century. The chief difference from medieval practice is a more rigid adherence to the diatonic notes of the scale. For example, in the medieval system, there is no practical difference between the Lydian scale and what we call the major scale, or between the Dorian and what we call the minor scale. In order to differentiate "modal flavor" from the chromatic flexibility of the upper half of the minor scale, for example, it became necessary to insist on always using the minor-seventh scale degree, especially in ascent to the tonic, and the sixth degree had to be always major to identify Dorian, and always minor to identify Aeolian. No such practice existed up to the 16th century, and indeed it was only with Glareanus in the mid-16th century that a theoretical difference was made between Dorian and Aeolian on the one hand, and Lydian and Ionian on the other. I shall see what I can do to clear this up (using reliable sources, of course).—