What Worship appears to be referring to is the actual physical distance between keys on the keyboard.
The group one family contain all white keys (C, G, F). If you were to measure the distances between the 1 - 3 and the 3 - 5, (center to center) you will find that the measurement is the same (although the musical distance is Maj 3 and min 3. In fact all of the chords that contain all white keys are the same. Thus you maintain the same hand shape for Dm, Em, Am, and B dim!
The second group has a black key in the middle (D, E, A). If you were to measure that, you would find that the 1 - 3 is physically larger than the 3 - 5. Also the 1 - 3 of each of these is physically larger than the 1 - 3 of the first group and the 3 - 5 is physically smaller than the 3 - 5 of the first group.
In this case, the minor chords with a black in the middle are NOT the same. Cm, Fm, Gm will not measure the same as D, E, A!
The same occurs in the third group (Db/C#, Eb, Ab). Since both Db/C#, and Ab still retain a white key in their minor version, there will also be a difference in physical distance. The distances are basically the same as the Black White Black group, but they fell different, particularly in relation to the thumb.
The "independent" are simply the chords with hand shapes that are unique to them. This includes Bb, B, Gb. Gb is physically the same measurements as the white key bunch, but will feel different. Also Ebm is physically the same as Gb.
This hand shape idea has to do with technic practice and is not necessarily meant to relate the chords by theory.
One way to practice your chords as arpeggios and such is to practice by group. Practice the C, F, G, Dm, Em, Am, B dim arpeggios in one session. Each one will support your learning on the others. The only difference will be their location on the keyboard in relation to where you sit.
Then work on other groups similarly.
After you have them by groups, then start mixing them up. You might try a group I followed by a group II (C D, F E, G A) etc. to help internalize the minute adjustments necessary between different hand shapes.
I will caution that, particularly with those that involve black keys, to begin by making sure that your finger is placed absolutely in the middle of the key. Get used to that feeling and you will be less likely to slip off or accidentally hit two keys.
