Like SimonTemplar, I too was surprised by the 11 card Celtic Cross offered here. However, unlike him, I have found the tarot.com version to be very satisfactory.
Maybe the intended reading pattern must be clear during shuffling (I use wash, box-riffle-riffle, box-riffle-riffle, strip-riffle-cut). But either way the cards read well for their intended variant and often wouldn't make as much sense with the other layout.
The Celtic Cross I am familiar with comes from Mary Greer's "Tarot for Your Self". She presents a 10 card spread, with comments on five traditions or sources. She also mentions that an additional card may be used for various functions. The variants as (a) Traditional (mostly Waite), (b) Jungian (mixture of sources), (c) Angeles Arrien, (d) Joanne Kowalski and (e) Others, including her own practice.
Greer's placement of Foundation, Recent Past, Higher Power and Near Future (to use Tarot.com terminology, she uses card numbers) is clockwise like Christine Payne-Towler's. SimonTemplar's account is interesting, placing these cards explicitly using a cross formation.
Greer makes much of the fact that the Celtic Cross uses 10 cards (numerologically significant). It is interesting that like tarot.com, SimonTemplar uses 11 cards. If the initial card is taken as a significator, the second as the situation and the third as what confounds/crosses the situation, his pattern follows Greer's "Traditional" and "Jungian" variants. Personally, I feel that "Self" is an easier label than significator and the central group-of-three between the vertical powers and horizontal times is intuitively appealing.
Regarding the "Self" position at the base of the staff, Greer's "Traditional" and "Jungian" readings are "Yourself as you see yourself" (traditional) and "The persona or mask" (Jungian). SimonTemplar's interpretation might be seen as a pure, possibly "fundamentalist" interpretation of these tendencies. To continue the analogy with Christian perspectives, Christine Payne-Towler is offering a "liberal" interpretation. As the "Self" is already represented, she is maybe emphasising those aspects of the self which the querent is hanging on to (and therefore potentially making the "cross" worse).
In my opinion, SimonTemplar's card 6 (soon future) and card 9 (soon outcome) are too similar. It may be that the contrast between the two may shed additional light, but such subtleties are surely beyond a computer generated system.
Using an Advice card in one these positions is surely essential. From a purely humane perspective, we all need something we can do about difficult situations. Cards that are not under direct human interpretation could easily be read in a mechanistic way. And a tarot system accessible from the privacy of your own home is likely to attract more querents who are vulnerable and afraid, not less. Without a modifier like "Advice", a reading could so easily crush rather than support.
As a final note, I would like to express my gratitude for the tone of Christine Payne-Towler's interpretations. It has been enlightening for me to see how it is possible to give informative, beneficial and even challenging readings while avoiding entirely any hint of negativity. As a Virgo, I tend to be critical and picky, so I am grateful to have her practical example to follow.