I'm not following your description of what you're reading on my webpage. It might help if you copied/pasted a few short specific statements in my webpage that are confusing you. For now, though, it might help for you to know that the basic results I give for the traveler's conclusions about the home twin's ageing during various portions of his trip are identical to the results of the analysis most commonly known as the "co-moving inertial frames" determination of the traveler's conclusions . So what is different about the description on my webpage primarily has to do with terminology, not results. The reason I have developed different terminology is that I've witnessed over many years a lot of confusion and mistakes being made by people trying to apply the conventional terminology. So my goal (undertaken many years ago) was to come up with terminology and variable names which reduce that confusion and those commonly made errors. And before that development of different terminology, I had derived an equation (now called "the CADO equation") which gets the same traveler's perspective normally obtained via the traveler's "line of simultaneity" on a Minkowski diagram, but in a much easier and less error-prone way. The CADO equation is also now expressed using the new terminology, again to minimize misunderstandings and errors.